text: K ∪ K Proof. The identity (3.6) trivially holds for d > dim( ) since all expressions on the ≤ K ∪ K 1 2 right and left are zero. Let d dim( ). The key tool now is the identity (2.3) for 1 2 Betti numbers. Substituting (3.4) in (2.3), we get that K − K − K K − K − K f ( ) γ ( ) γ ( ) + f ( ) γ ( ) γ ( ) = d 1 d 1 d+1 1 d 2 d 2 d+1 2 K ∩ K − K ∩ K − K ∩ K − f ( ) γ ( ) γ ( ) β(kerν )+ d 1 2 d 1 2 d+1 1 2 d K ∪ K − K ∪ K − K ∪ K − f ( ) γ ( ) γ ( ) β(kerν − ). d 1 2 d 1 2 d+1 1 2 d 1 The f terms cancel out due to inclusion-exclusion and hence the expression simpliﬁes to d K K K K K ∩ K K ∩ K (3.7) γ ( ) + γ ( ) + γ ( ) + γ ( ) =γ ( ) + γ ( ) + β(kerν )+ d 1 d+1 1 d 2 d+1 2 d 1 2 d+1 1 2 d K ∪ K K ∪ K γ ( ) + γ ( ) + β(kerν − ) d 1 2 d+1 1 2 d 1 K ∪ K · K K K ∪ K K ∩ K Firstly, let d = dim( ) and so γ ( ) = 0 for all , , , . 1 2 d+1 1 2 1 2 1 2 Furthermore, by exactness β(ker ν ) = 0. Substituting into (3.7), we obtain (3.6) for K ∪ K d d = dim( ). 1 2 From (3.7), it is easy to see that if (3.6) holds for d + 1, then it holds for d as well. Using K ∪ K (cid:3) this recursively and that (3.6) holds for d = dim( ), the proof follows. 1 2 ⊂ F Lemma 3.8 (Exchange property). Let S d be a spanning acycle of a simplicial complex K ∈ F \ ∈ and let σ d S. Then, for any d-face σ S such that σ is part of a d-cycle containing ∪ \ 1 1 σ, S σ σ is also a spanning acycle. 1 (cid:80) ∈ F \ K − ∪ K ∪ ∪ Proof. By Lemma 3.4, σ d S is a positive face w.r.t. d 1 S. So β ( S σ) = 1. ∪ ∈ − d Let C be the d-cycle in S σ. Clearly, σ C . So ∈ ∩ a ∂τ = ∂σ for some collection F− { } τ C S τ of non-zero valued coeﬃcients a . ∈ ∪ \τ Suppose that for σ S, S σ σ is not a spanning acycle. Then by Lemma 2.4, we K − ∪1 ∪ \ 1 K − ∪ (cid:80)∪ \ obtain that β ( d 1 S σ σ ) = β − ( d 1 S σ σ ) = 1. Let C be the d-cycle K − ∪ ∪d \ 1 (cid:54)⊂ d 1 1 1 ∈ in d 1 S σ σ . Clearly, C S as S is a spanning acycle. Hence σ C and we 1 1 (cid:48) ∈ F (cid:48) − 1 derive that for(cid:88)some collection of non-(cid:88)zero a , (cid:88)∈ ∩ a ∂τ = ∂σ. Setting a = 0 ∈ \ (cid:48) τ τ (C1 S) τ τ for τ C C and similarly for a , we derive that 1 τ (cid:48) − − − (a a )∂τ = a ∂τ a ∂τ = ∂σ ∂σ = 0. τ τ τ τ ∈ ∪ ∩ ∈ ∩ ∈ ∩ τ (C C1) S τ (C1 S) τ (C S) ∀ ∈ ∪ ∩ (cid:48) But since S is a spanning acycle, the above implies that τ (C C ) S, a = a and hence ∈ ∪ \ 1 τ τ C = C . So, we have that σ / C if S σ σ is not a spanning acycle. By contraposition, 1 ∈ 1 ∪ \ 1 (cid:3) we have that if σ C , then S σ σ is a spanning acycle. 1 1 20

text: K

text: K

text: Proof.

text: The

text: identity

text: trivially

text: holds

text: for

text: d

text: dim(

text: since

text: all

text: expressions

text: on

text: the

text: K

text: K

text: right

text: and

text: left

text: are

text: zero.

text: Let

text: d

text: dim(

text: The

text: key

text: tool

text: now

text: is

text: the

text: identity

text: for

text: Betti

text: numbers.

text: Substituting

text: in

text: we

text: get

text: that

text: K

text: K

text: K

text: K

text: K

text: K

text: f

text: γ

text: γ

text: f

text: γ

text: γ

text: d

text: d

text: d

text: d

text: K

text: K

text: K

text: K

text: K

text: K

text: f

text: γ

text: γ

text: β(kerν

text: d

text: d

text: d

text: K

text: K

text: K

text: K

text: K

text: K

text: f

text: γ

text: γ

text: β(kerν

text: d

text: d

text: d

text: The

text: f

text: terms

text: cancel

text: out

text: due

text: to

text: inclusion-exclusion

text: and

text: hence

text: the

text: expression

text: simpliﬁes

text: to

text: d

text: K

text: K

text: K

text: K

text: K

text: K

text: K

text: K

text: γ

text: γ

text: γ

text: γ

text: =γ

text: γ

text: β(kerν

text: d

text: d

text: d

text: d

text: K

text: K

text: K

text: K

text: γ

text: γ

text: β(kerν

text: d

text: d

text: K

text: K

text: K

text: K

text: K

text: K

text: K

text: K

text: Firstly,

text: let

text: d

text: dim(

text: and

text: so

text: γ

text: for

text: all

text: Furthermore,

text: by

text: exactness

text: β(ker

text: ν

text: Substituting

text: into

text: we

text: obtain

text: for

text: K

text: K

text: d

text: d

text: dim(

text: From

text: it

text: is

text: easy

text: to

text: see

text: that

text: if

text: holds

text: for

text: d

text: then

text: it

text: holds

text: for

text: d

text: as

text: well.

text: Using

text: K

text: K

text: this

text: recursively

text: and

text: that

text: holds

text: for

text: d

text: dim(

text: the

text: proof

text: follows.

text: F

text: Lemma

text: (Exchange

text: property).

text: Let

text: S

text: d

text: be

text: a

text: spanning

text: acycle

text: of

text: a

text: simplicial

text: complex

text: K

text: F

text: and

text: let

text: σ

text: d

text: S.

text: Then,

text: for

text: any

text: d-face

text: σ

text: S

text: such

text: that

text: σ

text: is

text: part

text: of

text: a

text: d-cycle

text: containing

text: σ,

text: S

text: σ

text: σ

text: is

text: also

text: a

text: spanning

text: acycle.

text: F

text: K

text: K

text: Proof.

text: By

text: Lemma

text: σ

text: d

text: S

text: is

text: a

text: positive

text: face

text: w.r.t.

text: d

text: S.

text: So

text: β

text: S

text: σ)

text: d

text: Let

text: C

text: be

text: the

text: d-cycle

text: in

text: S

text: σ.

text: Clearly,

text: σ

text: C

text: So

text: a

text: ∂τ

text: ∂σ

text: for

text: some

text: collection

text: F−

text: τ

text: C

text: S

text: τ

text: of

text: non-zero

text: valued

text: coeﬃcients

text: a

text: \τ

text: Suppose

text: that

text: for

text: σ

text: S,

text: S

text: σ

text: σ

text: is

text: not

text: a

text: spanning

text: acycle.

text: Then

text: by

text: Lemma

text: we

text: K

text: ∪1

text: K

text: obtain

text: that

text: β

text: d

text: S

text: σ

text: σ

text: β

text: d

text: S

text: σ

text: σ

text: Let

text: C

text: be

text: the

text: d-cycle

text: K

text: ∪d

text: d

text: in

text: d

text: S

text: σ

text: σ

text: Clearly,

text: C

text: S

text: as

text: S

text: is

text: a

text: spanning

text: acycle.

text: Hence

text: σ

text: C

text: and

text: we

text: F

text: derive

text: that

text: collection

text: of

text: a

text: a

text: ∂τ

text: ∂σ.

text: Setting

text: a

text: τ

text: τ

text: S)

text: τ

text: τ

text: for

text: τ

text: C

text: C

text: and

text: similarly

text: for

text: a

text: we

text: derive

text: that

text: τ

text: (a

text: a

text: )∂τ

text: a

text: ∂τ

text: a

text: ∂τ

text: ∂σ

text: ∂σ

text: τ

text: τ

text: τ

text: τ

text: τ

text: (C

text: S

text: τ

text: S)

text: τ

text: (C

text: S)

text: But

text: since

text: S

text: is

text: a

text: spanning

text: acycle,

text: the

text: above

text: implies

text: that

text: τ

text: (C

text: C

text: S,

text: a

text: a

text: and

text: hence

text: τ

text: τ

text: C

text: C

text: So,

text: we

text: have

text: that

text: σ

text: C

text: if

text: S

text: σ

text: σ

text: is

text: not

text: a

text: spanning

text: acycle.

text: By

text: contraposition,

text: we

text: have

text: that

text: if

text: σ

text: C

text: then

text: S

text: σ

text: σ

text: is

text: a

text: spanning

text: acycle.


text: K ≥ Deﬁnition 3.18 (Hypergraph connectivity). Let be a simplicial complex and let d 1. K F − Let GH( ) be the graph with vertex set V = d 1 and edge set d { ∈ F − × F − ∪ ∈ F } E = (σ , σ ) d 1 d 1 : σ σ d . 1 2 1 2 ⊂ F − K K A set S d 1 is said to be d-hypergraph connected in if S is connected in GH( ). The K K d complex is said to be d-hypergraph connected if GH( ) is connected. d When the dimension associated is clear, we shall speak of hypergraph connectivity omitting the dimension. We start with a crucial lemma for our hypergraph connectivity results. K K ∪ Lemma 3.19. Let be a d-complex. Suppose C is a d-acycle in such that C σ is a K ∪ K d-cycle (i.e., there is a d-chain in Z ( ) whose support is C σ and no d-chain in Z ( ) d K − ∪ d has support C ) then supp(∂C ) is d-hypergraph connected in d 1 C . As a consequence ⊂ K − ∪ supp(∂σ) supp(∂C ) is also d-hypergraph connected in d 1 C . (cid:80) (cid:80) ∈ Proof. Let σ and C be as assumed. This means that there exists ﬁeld coeﬃcients a , τ C ∈ τ (all non-trivial) such that ∂σ + ∈ a ∂τ = 0 but for all ﬁeld coeﬃcients b , τ C (not all (cid:54) τ C τ τ trivial) ∈ b ∂τ = 0. τ C τ K − ∪ F − F − If supp(∂C ) be not hypergraph connected in d 1 C then so is d 1. So, let d 1 = ∪ ∈ ∩ (cid:54) ∅ ∩ (cid:54) ∅ A B such that there is no τ C such that supp(∂τ) A = and supp(∂τ) B = . 1 1 1 ⊂1 Thus we can partition C into subsets A and B depending upon whether supp(∂τ) A or ⊂ ∪ ∩ ∅ 1 supp(∂τ) B respectively. Thus we get that C = A B with A B = . By deﬁnition 1 ∪ ∪ of C , we have that A σ and B σ are acyclic. We shall now derive a contradiction by ∪ showing that C σ is acyclic. K K − ∪ K K − ∪ K { ∈ K ⊂ } Deﬁne the complexes := d 2 A , = d 2 B and := τ : τ σ i.e., A 1 B K ∪ K 1K − σ K ∩ K K − σ viewed as a complete d-complex. Note that = d 1 and = d 2. Now K − ∪ ∪ A B A B to show d 1 C σ is acyclic, we use the Mayer-Vietoris long exact sequence (see Lemma K K ∪ ∪ K K K ∪ ∪ K 2.1). We set = A and = B . By deﬁnition of A , B , A, B, we K 1 K A σ 2 B σ K ∪ K 1K −1 ∪ ∪ have that and are both complexes. Further, we have that = d 1 C σ K ∩ K 1 K − 2∪ K 1 2 and = d 2 . Thus applying the Mayer-Vietoris long exact sequence, we have 1 2 σ · · · −→ K − ∪ K −→ K ∪ ∪ K ⊕ K ∪ ∪ K H ( d 2 ) H ( A ) H ( B ) d σ d A σ d B σ −→ K − ∪ ∪ K −→ K − ∪ K −→ · · · H ( d 1 C ) H − ( d 2 ) d σ d 1 σ K K K − ∪ ∪ Since , are acyclic, acyclicity of d 1 C σ follows from exactness of the above 1 2 K − ∪ K K − ∪ K sequence and the fact that H ( d 2 ) and H − ( d 2 ) are both trivial. The d σ d 1 σ F K − ∪ K triviality of the latter two homology groups follows from noting that d( d 2 ) = { } F − K − ∪ K F − K − ∪ K F − K K − ∪ K σ σ , d 1( d 2 ) = supp(∂σ), d 2( d 2 ) = d 2( ) i.e., d 2 is nothing − σ σ σ (cid:3) but (d 2)-skeleton of a complex with one d-simplex attached. 25

text: K

label: paragraph

text: Deﬁnition

text: (Hypergraph

text: connectivity).

text: Let

text: be

text: a

text: simplicial

text: complex

text: and

text: let

text: d

text: K

text: F

text: Let

text: GH(

text: be

text: the

text: graph

text: with

text: vertex

text: set

text: V

text: d

text: and

text: edge

text: set

text: d

text: F

text: F

text: F

text: E

text: (σ

text: σ

text: d

text: d

text: σ

text: σ

text: d

text: F

text: K

text: K

text: A

text: set

text: S

text: d

text: is

text: said

text: to

text: be

text: d-hypergraph

text: connected

text: in

text: if

text: S

text: is

text: connected

text: in

text: GH(

text: The

text: K

text: K

text: d

text: complex

text: is

text: said

text: to

text: be

text: d-hypergraph

text: connected

text: if

text: GH(

text: is

text: connected.

text: d

text: When

text: the

text: dimension

text: associated

text: is

text: clear,

text: we

text: shall

text: speak

text: of

text: hypergraph

text: connectivity

text: omitting

text: the

text: dimension.

text: We

text: start

text: with

text: a

text: crucial

text: lemma

text: for

text: our

text: hypergraph

text: connectivity

text: results.

text: K

text: K

text: Lemma

text: Let

text: be

text: a

text: d-complex.

text: Suppose

text: C

text: is

text: a

text: d-acycle

text: in

text: such

text: that

text: C

text: σ

text: is

text: a

text: K

text: K

text: d-cycle

text: (i.e.,

text: there

text: is

text: a

text: d-chain

text: in

text: Z

text: whose

text: support

text: is

text: C

text: σ

text: and

text: no

text: d-chain

text: in

text: Z

text: d

text: K

text: d

text: has

text: support

text: C

text: then

text: supp(∂C

text: is

text: d-hypergraph

text: connected

text: in

text: d

text: C

text: As

text: a

text: consequence

text: K

text: supp(∂σ)

text: supp(∂C

text: is

text: also

text: d-hypergraph

text: connected

text: in

text: d

text: C

text: Proof.

text: Let

text: σ

text: and

text: C

text: be

text: as

text: assumed.

text: This

text: means

text: that

text: there

text: exists

text: ﬁeld

text: coeﬃcients

text: a

text: τ

text: C

text: τ

text: (all

text: non-trivial)

text: such

text: that

text: ∂σ

text: a

text: ∂τ

text: but

text: for

text: all

text: ﬁeld

text: coeﬃcients

text: b

text: τ

text: C

text: (not

text: all

text: τ

text: C

text: τ

text: τ

text: trivial)

text: b

text: ∂τ

text: τ

text: C

text: τ

text: K

text: F

text: F

text: If

text: supp(∂C

text: be

text: not

text: hypergraph

text: connected

text: in

text: d

text: C

text: then

text: so

text: is

text: d

text: So,

text: let

text: d

text: A

text: B

text: such

text: that

text: there

text: is

text: no

text: τ

text: C

text: such

text: that

text: supp(∂τ)

text: A

text: and

text: supp(∂τ)

text: B

text: ⊂1

text: Thus

text: we

text: can

text: partition

text: C

text: into

text: subsets

text: A

text: and

text: B

text: depending

text: upon

text: whether

text: supp(∂τ)

text: A

text: or

text: supp(∂τ)

text: B

text: respectively.

text: Thus

text: we

text: get

text: that

text: C

text: A

text: B

text: with

text: A

text: B

text: By

text: deﬁnition

text: of

text: C

text: we

text: have

text: that

text: A

text: σ

text: and

text: B

text: σ

text: are

text: acyclic.

text: We

text: shall

text: now

text: derive

text: a

text: contradiction

text: by

text: showing

text: that

text: C

text: σ

text: is

text: acyclic.

text: K

text: K

text: K

text: K

text: K

text: K

text: Deﬁne

text: the

text: complexes

text: d

text: A

text: d

text: B

text: and

text: τ

text: τ

text: σ

text: i.e.,

text: A

text: B

text: K

text: K

text: σ

text: K

text: K

text: K

text: σ

text: viewed

text: as

text: a

text: complete

text: d-complex.

text: Note

text: that

text: d

text: and

text: d

text: Now

text: K

text: A

text: B

text: A

text: B

text: to

text: show

text: d

text: C

text: σ

text: is

text: acyclic,

text: we

text: use

text: the

text: Mayer-Vietoris

text: long

text: exact

text: sequence

text: (see

text: Lemma

text: K

text: K

text: K

text: K

text: K

text: K

text: We

text: set

text: A

text: and

text: B

text: By

text: deﬁnition

text: of

text: A

text: B

text: A,

text: B,

text: we

text: K

text: K

text: A

text: σ

text: B

text: σ

text: K

text: K

text: −1

text: have

text: that

text: and

text: are

text: both

text: complexes.

text: Further,

text: we

text: have

text: that

text: d

text: C

text: σ

text: K

text: K

text: K

text: 2∪

text: K

text: and

text: d

text: Thus

text: applying

text: the

text: Mayer-Vietoris

text: long

text: exact

text: sequence,

text: we

text: have

text: σ

text: K

text: K

text: K

text: K

text: K

text: K

text: H

text: d

text: H

text: A

text: H

text: B

text: d

text: σ

text: d

text: A

text: σ

text: d

text: B

text: σ

text: K

text: K

text: K

text: K

text: H

text: d

text: C

text: H

text: d

text: d

text: σ

text: d

text: σ

text: K

text: K

text: K

text: Since

text: are

text: acyclic,

text: acyclicity

text: of

text: d

text: C

text: σ

text: follows

text: from

text: exactness

text: of

text: the

text: above

text: K

text: K

text: K

text: K

text: sequence

text: and

text: the

text: fact

text: that

text: H

text: d

text: and

text: H

text: d

text: are

text: both

text: trivial.

text: The

text: d

text: σ

text: d

text: σ

text: F

text: K

text: K

text: triviality

text: of

text: the

text: latter

text: two

text: homology

text: groups

text: follows

text: from

text: noting

text: that

text: d(

text: d

text: F

text: K

text: K

text: F

text: K

text: K

text: F

text: K

text: K

text: K

text: σ

text: σ

text: d

text: d

text: supp(∂σ),

text: d

text: d

text: d

text: i.e.,

text: d

text: is

text: nothing

text: σ

text: σ

text: σ

text: but

text: (d

text: of

text: a

text: complex

text: with

text: one

text: d-simplex

text: attached.


text: from the fact that by deﬁnition the boundary chain of such a simplex reduces to 0. The − converse fact, that the existence of a pivot implies that it reduces the (k 1)-dimensional Betti number follows from Lemma 2.4. Furthermore, this correspondence highlights that all elements of cycle basis the persistence algorithm returns are of a special form - a linear combination of faces from the minimal spanning acycle and a simplex. K K D Theorem 3.23. Let be a weighted d-complex with β − ( ) = 0. Let be the point-set of H K d 1 death times in the persistence diagram of the − ( ) with the canonical ﬁltration induced by B d 1 H K the weights and let be the point-set of birth times in the persistence diagram of the ( ) d with the canonical ﬁltration induced by the weights. Then we have that D { ∈ } B { ∈ F \ } = w(σ) : σ M , = w(σ) : σ d M , K F K where M is a d-minimal spanning acycle of and d are the d-simplices of . D Proof. We shall only need to prove the result for death times . Because every d-simplex K − is either positive or negative with respect to (σ ) (see (2.7) and (2.8)). Further since by the incremental algorithm (Algorithm 1), negative simplices correspond to death times and D B positive simplices correspond to birth time ((2.9)), the result for implies that for . We again only consider the case when the ﬁltration values are unique and appeal to Remark 2.6 to complete the proof in the general case. Note that in the general case, we use the same total ordering for the incremental algorithm (Algorithm 1) generating death and birth times as well as the simplicial Kruskal’s algorithm (Algorithm 2). F By uniqueness of weights on d, we know that the Kruskal’s algorithm gives us the minimal spanning acycle M. Firstly, note that by the relation (2.5), the condition to add σ to S in (cid:40) ∪ Kruskal’s algorithm is equivalent to ∂(C (S)) ∂(C (S σ)). And similarly, we can observe d D d K − (cid:40) K − ∪ that the incremental algorithm adds c = w(σ) to if ∂(C ( (c ))) ∂(C ( (c ) σ)). ∈ K d d Let c be a value in the ﬁltration, i.e., there exists σ such that w(σ) = c. Let M(c) de- K ∩K note the acycle generated by Kruskal’s algorithm on (c), i.e., M(c) = M (c) and similarly, − we shall use the notation M(c ) as well. By the above discussion on Kruskal’s algorithm and K incremental algorithm, our proof is complete if we show that ∂(C (M(c))) = ∂(C ( (c))). ⊂ K d d Trivially, ∂(C (M(c))) ∂(C ( (c))) and we shall now show the other inclusion. d d ∈ F K \ Suppose the other inclusion does not hold, then there exists a τ d( (c)) M(c) such ∈ ∈ − that ∂τ / ∂(C (M(c))). Let w(τ) = b < c. Then clearly ∂τ / ∂(C (M(b ))) and so by d K − ∪ − d (2.5), τ will be a negative face with respect to d 1 M(b ). So, Kruskal’s algorithm would − ∈ have added σ to the acycle M(b ) contradicting the assumption that τ / M(c). Thus, we K (cid:3) have ∂(C (M(c))) = ∂(C ( (c))) and the proof is complete. d d The above result has strong applications for random complexes as will be seen in the next section. Firstly, we obtain [33, Theorem 1.1] (see (1.3)) as an easy corollary of our previous theorem and the representation (1.1). Further, we can easily prove a fundamental uniqueness 28

text: from

text: the

text: fact

text: that

text: by

text: deﬁnition

text: the

text: boundary

text: chain

text: of

text: such

text: a

text: simplex

text: reduces

text: to

text: The

text: converse

text: fact,

text: that

text: the

text: existence

text: of

text: a

text: pivot

text: implies

text: that

text: it

text: reduces

text: the

text: (k

text: Betti

text: number

text: follows

text: from

text: Lemma

text: Furthermore,

text: this

text: correspondence

text: highlights

text: that

text: all

text: elements

text: of

text: cycle

text: basis

text: the

text: persistence

text: algorithm

text: returns

text: are

text: of

text: a

text: special

text: form

text: a

text: linear

text: combination

text: of

text: faces

text: from

text: the

text: minimal

text: spanning

text: acycle

text: and

text: a

text: simplex.

text: K

text: K

text: D

text: Theorem

text: Let

text: be

text: a

text: weighted

text: d-complex

text: with

text: β

text: Let

text: be

text: the

text: point-set

text: of

text: H

text: K

text: d

text: death

text: times

text: in

text: the

text: persistence

text: diagram

text: of

text: the

text: with

text: the

text: canonical

text: ﬁltration

text: induced

text: by

text: B

text: d

text: H

text: K

text: the

text: weights

text: and

text: let

text: be

text: the

text: point-set

text: of

text: birth

text: times

text: in

text: the

text: persistence

text: diagram

text: of

text: the

text: d

text: with

text: the

text: canonical

text: ﬁltration

text: induced

text: by

text: the

text: weights.

text: Then

text: we

text: have

text: that

text: D

text: B

text: F

text: w(σ)

text: σ

text: M

text: w(σ)

text: σ

text: d

text: M

text: K

text: F

text: K

text: where

text: M

text: is

text: a

text: d-minimal

text: spanning

text: acycle

text: of

text: and

text: d

text: are

text: the

text: d-simplices

text: of

text: D

text: Proof.

text: We

text: shall

text: only

text: need

text: to

text: prove

text: the

text: result

text: for

text: death

text: times

text: Because

text: every

text: d-simplex

text: K

text: is

text: either

text: positive

text: or

text: negative

text: with

text: respect

text: to

text: (σ

text: (see

text: and

text: Further

text: since

text: by

text: the

text: incremental

text: algorithm

text: (Algorithm

text: negative

text: simplices

text: correspond

text: to

text: death

text: times

text: and

text: D

text: B

text: positive

text: simplices

text: correspond

text: to

text: birth

text: time

text: the

text: result

text: for

text: implies

text: that

text: for

text: We

text: again

text: only

text: consider

text: the

text: case

text: when

text: the

text: ﬁltration

text: values

text: are

text: unique

text: and

text: appeal

text: to

text: Remark

text: to

text: complete

text: the

text: proof

text: in

text: the

text: general

text: case.

text: Note

text: that

text: in

text: the

text: general

text: case,

text: we

text: use

text: the

text: same

text: total

text: ordering

text: for

text: the

text: incremental

text: algorithm

text: (Algorithm

text: generating

text: death

text: and

text: birth

text: times

text: as

text: well

text: as

text: the

text: simplicial

text: Kruskal’s

text: algorithm

text: (Algorithm

text: F

text: By

text: uniqueness

text: of

text: weights

text: on

text: d,

text: we

text: know

text: that

text: the

text: Kruskal’s

text: algorithm

text: gives

text: us

text: the

text: minimal

text: spanning

text: acycle

text: M.

text: Firstly,

text: note

text: that

text: by

text: the

text: relation

text: the

text: condition

text: to

text: add

text: σ

text: to

text: S

text: in

text: Kruskal’s

text: algorithm

text: is

text: equivalent

text: to

text: ∂(C

text: (S))

text: ∂(C

text: (S

text: σ)).

text: And

text: similarly,

text: we

text: can

text: observe

text: d

text: D

text: d

text: K

text: K

text: that

text: the

text: incremental

text: algorithm

text: adds

text: c

text: w(σ)

text: to

text: if

text: ∂(C

text: (c

text: ∂(C

text: (c

text: σ)).

text: K

text: d

text: d

text: Let

text: c

text: be

text: a

text: value

text: in

text: the

text: ﬁltration,

text: i.e.,

text: there

text: exists

text: σ

text: such

text: that

text: w(σ)

text: c.

text: Let

text: M(c)

text: de-

text: K

text: ∩K

text: note

text: the

text: acycle

text: generated

text: by

text: Kruskal’s

text: algorithm

text: on

text: (c),

text: i.e.,

text: M(c)

text: M

text: (c)

text: and

text: similarly,

text: we

text: shall

text: use

text: the

text: notation

text: M(c

text: as

text: well.

text: By

text: the

text: above

text: discussion

text: on

text: Kruskal’s

text: algorithm

text: and

text: K

text: incremental

text: algorithm,

text: our

text: proof

text: is

text: complete

text: if

text: we

text: show

text: that

text: ∂(C

text: (M(c)))

text: ∂(C

text: (c))).

text: K

text: d

text: d

text: Trivially,

text: ∂(C

text: (M(c)))

text: ∂(C

text: (c)))

text: and

text: we

text: shall

text: now

text: show

text: the

text: other

text: inclusion.

text: d

text: d

text: F

text: K

text: Suppose

text: the

text: other

text: inclusion

text: does

text: not

text: hold,

text: then

text: there

text: exists

text: a

text: τ

text: d(

text: (c))

text: M(c)

text: such

text: that

text: ∂τ

text: ∂(C

text: (M(c))).

text: Let

text: w(τ)

text: b

text: c.

text: Then

text: clearly

text: ∂τ

text: ∂(C

text: (M(b

text: and

text: so

text: by

text: d

text: K

text: d

text: τ

text: will

text: be

text: a

text: negative

text: face

text: with

text: respect

text: to

text: d

text: M(b

text: So,

text: Kruskal’s

text: algorithm

text: would

text: have

text: added

text: σ

text: to

text: the

text: acycle

text: M(b

text: contradicting

text: the

text: assumption

text: that

text: τ

text: M(c).

text: Thus,

text: we

text: K

text: have

text: ∂(C

text: (M(c)))

text: ∂(C

text: (c)))

text: and

text: the

text: proof

text: is

text: complete.

text: d

text: d

text: The

text: above

text: result

text: has

text: strong

text: applications

text: for

text: random

text: complexes

text: as

text: will

text: be

text: seen

text: in

text: the

text: next

text: section.

text: Firstly,

text: we

text: obtain

text: Theorem

text: (see

text: as

text: an

text: easy

text: corollary

text: of

text: our

text: previous

text: theorem

text: and

text: the

text: representation

text: Further,

text: we

text: can

text: easily

text: prove

text: a

text: fundamental

text: uniqueness


text: (cid:34) (cid:35) (cid:34) (cid:35) Hence M((cid:96)) can be r(cid:88)ewritten(cid:88)as (cid:89) (cid:88) (cid:89) n,d E (cid:96) (cid:48) | |E (cid:96) M((cid:96)) = 1(σ ; I) = [σσσ] 1(σ ; I) . n,d i i (cid:0) (cid:1) [σσσ]∈Γ σσσ(cid:48)(cid:48)(cid:48)∈I ((cid:96)):σσσ(cid:48)(cid:48)(cid:48)∼σσσ i=1 [σσσ]∈Γ i=1 (cid:0) (cid:1) n,d − − Counting the number of ways of choosing (cid:96) distinct (d 1) faces from a total of n , we d | | ∈ ⊂ | | | | have I ((cid:96)) = (cid:96)! (nd) . Since, for each [σσσ] Γ, [σσσ] I ((cid:96)), we write [[[σσσ]]] = c ([σσσ]) I ((cid:96)) , for n,d ∈ (cid:96) n,d n n,d some c ([σσσ]) [0, 1]. Clearly (cid:88) n (4.6) c ([σσσ]) = 1. n ∈ [σσσ] Γ (cid:18)(cid:0) (cid:1)(cid:19) (cid:34) (cid:35) (cid:88) (cid:89) Hence it follows that n (cid:96) E M((cid:96)) = c ([σσσ])(cid:96)! d 1(σ ; I) . n,d n (cid:96) i ∈ [σσσ] Γ i=1 ∈ F(cid:88)− U (cid:88) Since for every σ d 1( ), n,d m m ∞ − ∞ 1(σ; I) = 1(σ; (a − , a ]) = (1(σ; (a − , )) 1(σ; (a , ))), 2j 1 2j 2j 1 2j j=1 j=1 ≡ ∈ it follows that for(cid:89)any σσσ (σ , . . . , σ )(cid:88)I ((cid:96)), (cid:89) 1 (cid:96) n,d (cid:96) (cid:96) − ∞ 1(σ ; I) = ( 1)αi+11(σ ; (a , )), i i αi ∈{ } i=1 (α1,...,α(cid:96)) 1,...,2m (cid:96) i=1 ⊆ R where, for any R , R(cid:96) denotes the cartesia(cid:18)n (cid:0)pr(cid:1)o(cid:19)duct of R take(cid:34)n (cid:96) times. Hence, (cid:35) (cid:88) (cid:88) (cid:89) (cid:80) n − E (cid:96) ∞ (4.7) Mn((cid:96),d) = cn([σσσ]) (cid:96)! (cid:96)d ( 1) i αi+l 1(σi; (aαi, )) . ∈ ∈{ } [σσσ] Γ (α1,...,α(cid:96)) 1,...,2m (cid:96) i=1 ∈ F − U ∈ R By the scaling of C(σ), note that, for any σ d 1( ) and any a , n,d − ∞ a + d log n log(d!) 1(σ, (a, )) = 1[C(σ) > ] n (cid:18)(cid:0) (cid:1)(cid:19) (cid:34) (cid:35) (cid:18) (cid:19) (cid:89) (cid:89) Combining this with (4.1) and (2) from Deﬁnition 4.1, observe that − − n E (cid:96) ∞ ∼ nd(cid:96) (cid:96) − a + d log n log(d!) n κi (cid:96)! d 1(σ ; (a , )) 1 αi . (cid:96) i αi (d!)(cid:96) n i=1 i=1 34 ##LTLine## ##LTLine## ##LTLine##

text: Hence

text: can

text: be

text: n,d

text: E

text: |E

text: I)

text: [σσσ]

text: I)

text: n,d

text: i

text: i

text: [σσσ]∈Γ

text: [σσσ]∈Γ

text: n,d

text: Counting

text: the

text: number

text: of

text: ways

text: of

text: choosing

text: distinct

text: (d

text: faces

text: from

text: a

text: total

text: of

text: n

text: we

text: d

text: have

text: I

text: (nd)

text: Since,

text: for

text: each

text: [σσσ]

text: Γ,

text: [σσσ]

text: I

text: we

text: write

text: [[[σσσ]]]

text: c

text: ([σσσ])

text: I

text: for

text: n,d

text: n,d

text: n

text: n,d

text: some

text: c

text: ([σσσ])

text: Clearly

text: n

text: c

text: ([σσσ])

text: n

text: [σσσ]

text: Γ

text: Hence

text: it

text: follows

text: that

text: n

text: E

text: c

text: d

text: I)

text: n,d

text: n

text: i

text: [σσσ]

text: Γ

text: U

text: Since

text: for

text: every

text: σ

text: d

text: n,d

text: m

text: m

text: I)

text: (a

text: a

text: (a

text: (a

text: it

text: follows

text: that

text: σσσ

text: (σ

text: σ

text: n,d

text: I)

text: (a

text: i

text: i

text: αi

text: R

text: where,

text: for

text: any

text: R

text: denotes

text: the

text: of

text: R

text: times.

text: Hence,

text: n

text: E

text: cn([σσσ])

text: i

text: αi+l

text: (aαi,

text: [σσσ]

text: Γ

text: F

text: U

text: R

text: By

text: the

text: scaling

text: of

text: C(σ),

text: note

text: that,

text: for

text: any

text: σ

text: d

text: and

text: any

text: a

text: n,d

text: a

text: d

text: log

text: n

text: log(d!)

text: (a,

text: n

text: Combining

text: this

text: with

text: and

text: from

text: Deﬁnition

text: observe

text: that

text: n

text: E

text: a

text: d

text: log

text: n

text: log(d!)

text: n

text: κi

text: d

text: (a

text: αi

text: i

text: αi

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48) − − − y (x) p(x)y(x) = q(x)h[x τ(x)] + s(x) if x τ(x) < a (4) y(a) = h(a) (5) { } Discretize [a, b] into a set N scattered nodes ξ = x , j = 1...N (with x = a and − j 1 x = b), and consider as well the outside point x = a λ, λ > 0. We seek an approximate N 0 solution to (3(cid:88))-(5) in the form of an expansion of N + 1 RBFs φ (r): j j=N (cid:107) − (cid:107) y(x) = α φ( x x ) (6) j j j=0 The addition of an RBF at x allows to enforce both the initial condition and the DDE 0 at x = a, thus contributing to the accuracy (this is the PDECB strategy discussed in [11]). Once the coeﬃcients α are available, the approximate RBF solution can be reconstructed j anywhere in [a, b]. In order to solve for the coeﬃcients, (3)-(5) are enforced over (6) on a set of collocation N nodes, usually ξ. Notice that no equation is collocated on x , but two of 0 them a(cid:88)re on x = a. For i = 1, . . . , N, this leads to the linear system of dimension N + 1 1 j=N { (cid:48) − − || − − || } − φ (r ) p(x )φ (r ) q(x )φ ( x τ(x ) x ) = s(x ) if x τ(x ) > a (7) j ij i j ij i j i i j i i i j=0 (cid:88) j=N { (cid:48) − } − − ≤ φ (r ) p(x )φ (r ) α = q(x )h[x τ(x )] + s(x ) if x τ(x ) a (8) j ij i j ij j i i i i i i j=0 (cid:88) j=N α φ (r ) = h(a) if x = a (9) j j ij i j=0 || − || where rij =(cid:113) xi xj . In the remainder of this paper we will restrict ourselves to the well-tested Hardy’s multiquadric (MQ), (cid:107) − (cid:107) φ (r ) = x x 2 +c2 (10) j j j j whose deriva(cid:113)tive is − (cid:48) x x φ (r ) = j (11) j j (cid:107) − (cid:107) x x 2 +c2 j j as the RBF of choice. The shape of the MQ depends on the free parameter c (hence j the name of shape parameter for it). The fact that the MQ has global support leads to fully populated matrices. It is a hallmark of Kansa’s method that the best accuracy can only be obtained at the expense of extreme ill-conditioning, as will be discussed next. In order to improve stability, the direct inversion of the linear system (7)-(9) has been replaced by the use of Penrose’s pseudoinverse. 2.1 Choosing the shape parameters c j Although the accuracy of the interpolant (6) is largely inﬂuenced by the values c , j = j 0, . . . , N + 1, theoretical results regarding the choice of an ’optimal’ set of values are not yet available, and heuristic rules must be used instead, which mostly address the homogeneous case cj = c. In this case, the convergence rate of the error of the interpolant (6) ha√s been proven to go as Λc/h in interpolation problems [22], and has been shown to obey Λ c/h in elliptic PDEs [6], where 0 < Λ < 1 and h is the distance between nodes. Therefore, the 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: y

text: (x)

text: p(x)y(x)

text: q(x)h[x

text: τ(x)]

text: s(x)

text: if

text: x

text: τ(x)

text: a

text: y(a)

text: h(a)

text: Discretize

text: [a,

text: b]

text: into

text: a

text: set

text: N

text: scattered

text: nodes

text: ξ

text: x

text: j

text: (with

text: x

text: a

text: and

text: j

text: x

text: b),

text: and

text: consider

text: as

text: well

text: the

text: outside

text: point

text: x

text: a

text: λ,

text: λ

text: We

text: seek

text: an

text: approximate

text: N

text: solution

text: to

text: in

text: the

text: form

text: of

text: an

text: expansion

text: of

text: N

text: RBFs

text: φ

text: (r):

text: j

text: j=N

text: y(x)

text: α

text: φ(

text: x

text: x

text: j

text: j

text: The

text: addition

text: of

text: an

text: RBF

text: at

text: x

text: allows

text: to

text: enforce

text: both

text: the

text: initial

text: condition

text: and

text: the

text: DDE

text: at

text: x

text: a,

text: thus

text: contributing

text: to

text: the

text: accuracy

text: (this

text: is

text: the

text: PDECB

text: strategy

text: discussed

text: in

text: Once

text: the

text: coeﬃcients

text: α

text: are

text: available,

text: the

text: approximate

text: RBF

text: solution

text: can

text: be

text: reconstructed

text: j

text: anywhere

text: in

text: [a,

text: b].

text: In

text: order

text: to

text: solve

text: for

text: the

text: coeﬃcients,

text: are

text: enforced

text: over

text: on

text: a

text: set

text: of

text: collocation

text: N

text: nodes,

text: usually

text: ξ.

text: Notice

text: that

text: no

text: equation

text: is

text: collocated

text: on

text: x

text: but

text: two

text: of

text: them

text: on

text: x

text: a.

text: For

text: i

text: N,

text: this

text: leads

text: to

text: the

text: linear

text: system

text: of

text: dimension

text: N

text: j=N

text: φ

text: (r

text: p(x

text: )φ

text: (r

text: q(x

text: )φ

text: x

text: τ(x

text: x

text: s(x

text: if

text: x

text: τ(x

text: a

text: j

text: ij

text: i

text: j

text: ij

text: i

text: j

text: i

text: i

text: j

text: i

text: i

text: i

text: j=N

text: φ

text: (r

text: p(x

text: )φ

text: (r

text: α

text: q(x

text: )h[x

text: τ(x

text: s(x

text: if

text: x

text: τ(x

text: a

text: j

text: ij

text: i

text: j

text: ij

text: j

text: i

text: i

text: i

text: i

text: i

text: i

text: j=N

text: α

text: φ

text: (r

text: h(a)

text: if

text: x

text: a

text: j

text: j

text: ij

text: i

text: where

text: rij

text: xi

text: xj

text: In

text: the

text: remainder

text: of

text: this

text: paper

text: we

text: will

text: restrict

text: ourselves

text: to

text: the

text: well-tested

text: Hardy’s

text: multiquadric

text: (MQ),

text: φ

text: (r

text: x

text: x

text: j

text: j

text: j

text: j

text: whose

text: is

text: x

text: x

text: φ

text: (r

text: j

text: j

text: j

text: x

text: x

text: j

text: j

text: as

text: the

text: RBF

text: of

text: choice.

text: The

text: shape

text: of

text: the

text: MQ

text: depends

text: on

text: the

text: free

text: parameter

text: c

text: (hence

text: j

text: the

text: name

text: of

text: shape

text: parameter

text: for

text: it).

text: The

text: fact

text: that

text: the

text: MQ

text: has

text: global

text: support

text: leads

text: to

text: fully

text: populated

text: matrices.

text: It

text: is

text: a

text: hallmark

text: of

text: Kansa’s

text: method

text: that

text: the

text: best

text: accuracy

text: can

text: only

text: be

text: obtained

text: at

text: the

text: expense

text: of

text: extreme

text: ill-conditioning,

text: as

text: will

text: be

text: discussed

text: next.

text: In

text: order

text: to

text: improve

text: stability,

text: the

text: direct

text: inversion

text: of

text: the

text: linear

text: system

text: has

text: been

text: replaced

text: by

text: the

text: use

text: of

text: Penrose’s

text: pseudoinverse.

text: Choosing

text: the

text: shape

text: parameters

text: c

text: j

text: Although

text: the

text: accuracy

text: of

text: the

text: interpolant

text: is

text: largely

text: inﬂuenced

text: by

text: the

text: values

text: c

text: j

text: j

text: N

text: theoretical

text: results

text: regarding

text: the

text: choice

text: of

text: an

text: ’optimal’

text: set

text: of

text: values

text: are

text: not

text: yet

text: available,

text: and

text: heuristic

text: rules

text: must

text: be

text: used

text: instead,

text: which

text: mostly

text: address

text: the

text: homogeneous

text: case

text: cj

text: c.

text: In

text: this

text: case,

text: the

text: convergence

text: rate

text: of

text: the

text: error

text: of

text: the

text: interpolant

text: ha√s

text: been

text: proven

text: to

text: go

text: as

text: Λc/h

text: in

text: interpolation

text: problems

text: and

text: has

text: been

text: shown

text: to

text: obey

text: Λ

text: c/h

text: in

text: elliptic

text: PDEs

text: where

text: Λ

text: and

text: h

text: is

text: the

text: distance

text: between

text: nodes.

text: Therefore,

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [16] K. Ito, H.T. Tran, and A. Manitius, A Fully-Discrete Spectral Method for Delay Diﬀer- ential Equations, SIAM J. Numerical Analysis, 28, 1121-1140 (1991). [17] J.H. Jung, A note on the Gibbs phenomenon with multiquadric radial basis functions, Applied Numerical Mathematics 57, Issue 2, 213-229 (2007). [18] E. J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational ﬂuid-dynamics. I. Surface approximations and partial derivative esti- mates, Comput. Math. Appls. 19, 127-145 (1990). [19] E. J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational ﬂuid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial diﬀerential equations, Comput. Math. Appls. 19, 147-161 (1990). [20] Kansa, E.J. and Carlson, R.E., Improved accuracy of multiquadric interpolation using variable shape parameters, Comput. Math. Appl. 24, 99-120 (1992). [21] S. Karimi Vanani and A. Aminataei, Multiquadric approximation scheme on the numer- ical solution of delay diﬀerential systems of neutral type, Mathematical and Computer Modelling, In Press (2008). [22] W.R. Madych, Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Appl. 24, 121-38 (1992). [23] S.B. Norkin and L.E. El’sgol’ts, Introduction to the Theory and Applications of Diﬀer- ential Equations with Deviating Arguments, Math. in Sci. and Eng. 105, 44-45 (1973). [24] C.A.H. Paul, A test set of functional diﬀerential equations, Technical Report 243, Uni- versity of Manchester (1992). [25] M.J.D. Powell, A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equa- tions, Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7 (1970). [26] S. Rippa An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math. 11(2-3), 193-210 (1999). [27] R. Schaback, Error estimates and condition numbers for radial basis function interpola- tion, Adv. Comput. Math. 3, 251-264 (1995). [28] L.F. Shampine and S. Thompsom, Solving DDEs in Matlab, Appl. Numer. Math., 37, 441-458 (2001). [29] L.F. Shampine, Solving ODEs and DDEs with residual control, Appl. Numer. Math., 52, 113-127 (2005). [30] J. Wertz, E. J. Kansa and L. Ling, The role of the multiquadric shape parameters in solv- ing elliptic partial diﬀerential equations, Computers and Mathematics with Applications 51(8), 1335-1348 (2006). 12

label: reference

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text: Delay

text: Diﬀer-

text: ential

text: Equations,

text: SIAM

text: J.

text: Numerical

text: Analysis,

text: J.H.

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text: A

text: note

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text: phenomenon

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text: Numerical

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text: computational

text: ﬂuid-dynamics.

text: II.

text: Solutions

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text: parabolic,

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text: Multiquadric

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text: numer-

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text: and

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text: of

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text: ential

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text: with

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text: Math.

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text: Powell,

text: A

text: Fortran

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text: Solving

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text: Nonlinear

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text: Numerical

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text: Shampine,

text: Solving

text: ODEs

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text: with

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text: E.

text: J.

text: Kansa

text: and

text: L.

text: Ling,

text: The

text: role

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text: the

text: multiquadric

text: shape

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text: in

text: solv-

text: ing

text: elliptic

text: partial

text: diﬀerential

text: equations,

text: Computers

text: and

text: Mathematics

text: with

text: Applications


text: [8] Jiashi Feng, Huan Xu, Shie Mannor, and Shuicheng Yan. Online pca for contaminated data. In NIPS, pages 764–772, 2013. [9] Jiashi Feng, Huan Xu, and Shuicheng Yan. Online robust pca via stochastic optimiza- tion. In NIPS, 2013. [10] N. Guan, D. Tao, Z. Luo, and B. Yuan. Online nonnegative matrix factorization with robust stochastic approximation. Neural Networks and Learning Systems, IEEE Transactions on, 23(7):1087–1099, 2012. [11] Daniel Hsu and Sivan Sabato. Loss minimization and parameter estimation with heavy tails. arXiv preprint arXiv:1307.1827, 2013. [12] Peter J Huber. Robust statistics. Springer, 2011. [13] Yangqing Jia, Evan Shelhamer, Jeﬀ Donahue, Sergey Karayev, Jonathan Long, Ross Girshick, Sergio Guadarrama, and Trevor Darrell. Caﬀe: Convolutional architecture for fast feature embedding. arXiv preprint arXiv:1408.5093, 2014. [14] Matthieu Lerasle and Roberto I Oliveira. Robust empirical mean estimators. arXiv preprint arXiv:1112.3914, 2011. [15] Julien Mairal, Francis Bach, Jean Ponce, and Guillermo Sapiro. Online dictionary learning for sparse coding. In ICML, pages 689–696, 2009. [16] Ricardo A Maronna and V´ıctor J Yohai. Robust estimation of multivariate location and scatter. Encyclopedia of Statistical Sciences, 1998. [17] Stanislav Minsker. Geometric median and robust estimation in banach spaces. arXiv preprint arXiv:1308.1334, 2013. [18] Angelia Nedic, DP Bertsekas, and AE Ozdaglar. Convex analysis and optimization. Athena Scientiﬁc, 2003. [19] Alexander Rakhlin, Ohad Shamir, and Karthik Sridharan. Making gradient descent optimal for strongly convex stochastic optimization. arXiv preprint arXiv:1109.5647, 2011. [20] Roman Vershynin. How close is the sample covariance matrix to the actual covariance matrix? Journal of Theoretical Probability, 25(3):655–686, 2012. [21] Huan Xu, C. Caramanis, and S. Mannor. Outlier-robust pca: The high-dimensional case. Information Theory, IEEE Transactions on, 59(1):546–572, 2013. 18

text: Jiashi

text: Feng,

text: Huan

text: Xu,

text: Shie

text: Mannor,

text: and

text: Shuicheng

text: Yan.

text: Online

text: pca

text: for

text: contaminated

text: data.

text: In

text: NIPS,

text: pages

text: Jiashi

text: Feng,

text: Huan

text: Xu,

text: and

text: Shuicheng

text: Yan.

text: Online

text: robust

text: pca

text: via

text: stochastic

text: optimiza-

text: tion.

text: In

text: NIPS,

text: N.

text: Guan,

text: D.

text: Tao,

text: Z.

text: Luo,

text: and

text: B.

text: Yuan.

text: Online

text: nonnegative

text: matrix

text: factorization

text: with

text: robust

text: stochastic

text: approximation.

text: Neural

text: Networks

text: and

text: Learning

text: Systems,

text: IEEE

text: Transactions

text: on,

text: Daniel

text: Hsu

text: and

text: Sivan

text: Sabato.

text: Loss

text: minimization

text: and

text: parameter

text: estimation

text: with

text: heavy

text: tails.

text: arXiv

text: preprint

text: Peter

text: J

text: Huber.

text: Robust

text: statistics.

text: Springer,

text: Yangqing

text: Jia,

text: Evan

text: Shelhamer,

text: Jeﬀ

text: Donahue,

text: Sergey

text: Karayev,

text: Jonathan

text: Long,

text: Ross

text: Girshick,

text: Sergio

text: Guadarrama,

text: and

text: Trevor

text: Darrell.

text: Caﬀe:

text: Convolutional

text: architecture

text: for

text: fast

text: feature

text: embedding.

text: arXiv

text: preprint

text: Matthieu

text: Lerasle

text: and

text: Roberto

text: I

text: Oliveira.

text: Robust

text: empirical

text: mean

text: estimators.

text: arXiv

text: preprint

text: Julien

text: Mairal,

text: Francis

text: Bach,

text: Jean

text: Ponce,

text: and

text: Guillermo

text: Sapiro.

text: Online

text: dictionary

text: learning

text: for

text: sparse

text: coding.

text: In

text: ICML,

text: pages

text: Ricardo

text: A

text: Maronna

text: and

text: V´ıctor

text: J

text: Yohai.

text: Robust

text: estimation

text: of

text: multivariate

text: location

text: and

text: scatter.

text: Encyclopedia

text: of

text: Statistical

text: Sciences,

text: Stanislav

text: Minsker.

text: Geometric

text: median

text: and

text: robust

text: estimation

text: in

text: banach

text: spaces.

text: arXiv

text: preprint

text: Angelia

text: Nedic,

text: DP

text: Bertsekas,

text: and

text: AE

text: Ozdaglar.

text: Convex

text: analysis

text: and

text: optimization.

text: Athena

text: Scientiﬁc,

text: Alexander

text: Rakhlin,

text: Ohad

text: Shamir,

text: and

text: Karthik

text: Sridharan.

text: Making

text: gradient

text: descent

text: optimal

text: for

text: strongly

text: convex

text: stochastic

text: optimization.

text: arXiv

text: preprint

text: Roman

text: Vershynin.

text: How

text: close

text: is

text: the

text: sample

text: covariance

text: matrix

text: to

text: the

text: actual

text: covariance

text: matrix?

text: Journal

text: of

text: Theoretical

text: Probability,

text: Huan

text: Xu,

text: C.

text: Caramanis,

text: and

text: S.

text: Mannor.

text: Outlier-robust

text: pca:

text: The

text: high-dimensional

text: case.

text: Information

text: Theory,

text: IEEE

text: Transactions

text: on,


text: or communication errors, as demonstrated in the experiments. In short, we make following contributions in this work. First, we develop an outlier ro- bust online learning framework which is the ﬁrst one with provable robustness to a constant fraction of outliers. Secondly, we introduce two concrete online robust learning approaches, one for unsupervised learning and the other for supervised learning. Other examples can be developed in a similar way easily. Finally, we also present the application of the ORL approach to distributed learning setting which is equally attractive for learning from large scale data. 2 Preliminaries 2.1 Problem Set-up X X ∪ X { } ∪ We consider a set of N=n+n observation samples = I O = x , . . . , x { } ⊂ R 1 X 1 n x , . . . , x p, which contains a mixture of n authentic samples I and n outliers X n+1 n+n1 1 O. The authentic samples are generated according to an underlying model (i.e., the ground ∈ truth) parameterized by θ(cid:63) Θ. The target of a statistical learning procedure is to estimate X the model parameter θ(cid:63) according to the provided observations . Throughout the paper, R we assume the authentic samples are sub-Gaussian random vectors in p, which thus satisfy that P(|(cid:104)x, u(cid:105)| > t) ≤ 2e−t2/L2 for t > 0 and u ∈ Sp−1, (1) − for some L. Here Sp 1 denotes the unit sphere. In this work, we focus on the case where a constant fraction of the observations are out- (cid:44) liers, and we use λ n /N to denote this outlier fraction. In the context of online learning, 1 (cid:98) samples are provided in a sequence of T mini batches, each of which contains b = (n + (cid:99) {X X } { } n )/T observations. Denote the sequence as , . . . , = x , . . . , x , x , . . . , x . 1 1 T 1 b b+1 Tb The target of online statistical learning is to estimate the parameter θ(cid:63), only based on the observations revealed so far. 2.2 Geometric Median We ﬁrst introduce the geometric median here—a core concept underlying the median ﬁltering procedure that is important for developing the proposed online robust learning approach. ∈ Deﬁnition 1 (Geometric Median). Given a ﬁnite collection of i.i.d. estimates θ , . . . , θ 1 T Θ, their geometric median is the point which minimizes the total (cid:96) distance to all the given 1 estimates, i.e., (cid:88) (cid:98) T (cid:107) − (cid:107) θ = median(θ , . . . , θ ) := arg min θ θ . (2) 1 T ∈ j θ Θ j=1 3

text: or

text: communication

text: errors,

text: as

text: demonstrated

text: in

text: the

text: experiments.

text: In

text: short,

text: we

text: make

text: following

text: contributions

text: in

text: this

text: work.

text: First,

text: we

text: develop

text: an

text: outlier

text: ro-

text: bust

text: online

text: learning

text: framework

text: which

text: is

text: the

text: ﬁrst

text: one

text: with

text: provable

text: robustness

text: to

text: a

text: constant

text: fraction

text: of

text: outliers.

text: Secondly,

text: we

text: introduce

text: two

text: concrete

text: online

text: robust

text: learning

text: approaches,

text: one

text: for

text: unsupervised

text: learning

text: and

text: the

text: other

text: for

text: supervised

text: learning.

text: Other

text: examples

text: can

text: be

text: developed

text: in

text: a

text: similar

text: way

text: easily.

text: Finally,

text: we

text: also

text: present

text: the

text: application

text: of

text: the

text: ORL

text: approach

text: to

text: distributed

text: learning

text: setting

text: which

text: is

text: equally

text: attractive

text: for

text: learning

text: from

text: large

text: scale

text: data.

label: section

text: Preliminaries

text: Problem

text: Set-up

text: X

text: X

text: X

text: We

text: consider

text: a

text: set

text: of

text: N=n+n

text: observation

text: samples

text: I

text: O

text: x

text: x

text: R

text: X

text: n

text: x

text: x

text: p,

text: which

text: contains

text: a

text: mixture

text: of

text: n

text: authentic

text: samples

text: I

text: and

text: n

text: outliers

text: X

text: O.

text: The

text: authentic

text: samples

text: are

text: generated

text: according

text: to

text: an

text: underlying

text: model

text: (i.e.,

text: the

text: ground

text: truth)

text: parameterized

text: by

text: Θ.

text: The

text: target

text: of

text: a

text: statistical

text: learning

text: procedure

text: is

text: to

text: estimate

text: X

text: the

text: model

text: parameter

text: according

text: to

text: the

text: provided

text: observations

text: Throughout

text: the

text: paper,

text: R

text: we

text: assume

text: the

text: authentic

text: samples

text: are

text: sub-Gaussian

text: random

text: vectors

text: in

text: p,

text: which

text: thus

text: satisfy

text: that

text: t)

text: for

text: t

text: and

text: u

text: for

text: some

text: L.

text: Here

text: Sp

text: denotes

text: the

text: unit

text: sphere.

text: In

text: this

text: work,

text: we

text: focus

text: on

text: the

text: case

text: where

text: a

text: constant

text: fraction

text: of

text: the

text: observations

text: are

text: out-

text: liers,

text: and

text: we

text: use

text: λ

text: n

text: /N

text: to

text: denote

text: this

text: outlier

text: fraction.

text: In

text: the

text: context

text: of

text: online

text: learning,

text: samples

text: are

text: provided

text: in

text: a

text: sequence

text: of

text: T

text: mini

text: batches,

text: each

text: of

text: which

text: contains

text: b

text: (n

text: {X

text: X

text: n

text: )/T

text: observations.

text: Denote

text: the

text: sequence

text: as

text: x

text: x

text: x

text: x

text: T

text: b

text: Tb

text: The

text: target

text: of

text: online

text: statistical

text: learning

text: is

text: to

text: estimate

text: the

text: parameter

text: only

text: based

text: on

text: the

text: observations

text: revealed

text: so

text: far.

text: Geometric

text: Median

text: We

text: ﬁrst

text: introduce

text: the

text: geometric

text: median

text: here—a

text: core

text: concept

text: underlying

text: the

text: median

text: ﬁltering

text: procedure

text: that

text: is

text: important

text: for

text: developing

text: the

text: proposed

text: online

text: robust

text: learning

text: approach.

text: Deﬁnition

text: (Geometric

text: Median).

text: Given

text: a

text: ﬁnite

text: collection

text: of

text: i.i.d.

text: estimates

text: θ

text: θ

text: T

text: Θ,

text: their

text: geometric

text: median

text: is

text: the

text: point

text: which

text: minimizes

text: the

text: total

text: distance

text: to

text: all

text: the

text: given

text: estimates,

text: i.e.,

text: T

text: θ

text: median(θ

text: θ

text: arg

text: min

text: θ

text: θ

text: T

text: j

text: θ

text: Θ


text: Algorithm 1 The ORL Approach X X Input: Mini-bat(cid:98)ch sequence , . . . , , convexity parameter c , robust learning proce- 1 T a dure parameter ν. Initialization: θ = 0. 0 for t = 1, . . . , T do (cid:98) X Call the robust learnin(cid:98)g procedure (cid:98)θ = RL( , ν); (cid:107) − t(cid:107) t Compute weight w = 2η / θ − θ with η = 1/c t. t t t−1 t t a Update the estimate: θ(cid:98)= (1 w )θ − + w θ . t t t 1 t t+1 end for Output: Final estimate θ . T (cid:98) estimate θ , ORL computes the gradient at point θ of the empirical function G in Eqn. (3) t T evaluated only at θ : (cid:98) t+1 (cid:98) − (cid:44) ∂G (θ; θ ) 2(θ θ ) g(θ; θ ) T t+1 = (cid:107) − t+1(cid:107) . (4) t+1 (cid:98) ∂θ θ θ t+1 Then ORL update(cid:98)s the es(cid:98)timate(cid:98)θt(cid:98)by following ﬁlteri(cid:98)ng: ← − − θ θ η g(θ ; θ ) = (1 w )θ + w θ . (5) t+1 t t t t+1 t t t t+1 He(cid:98)re η is a predeﬁned step size parameter which usually takes the form of 1/c t with t a a constant c cha(cid:98)racterizing convexity of the empirical function to optimize. Besides, w = (cid:107) − a(cid:107) t 2η / θ θ controls contribution of each new estimate θ conservatively in updating the t t t+1 t+1 global estimate θ. Details of ORL are provided in Algorithm 1. Another level of ﬁltering is important. Certain mini-batches may contain overwhelming outliers. Therefore, even though a robust learning procedure is employed on each mini-batch, the resulted estimate cannot be guaranteed to be accurate. In fact, a mini-batch containing over 50% outliers would corrupt any robust learning procedure—the resulted estimate can be arbitrarily bad and breakdown the overall online learning. To address this critical issue, ORL performs another level of online learning for updating the “global” estimate with adaptive weights for the new estimate and “ﬁlters out” possibly bad estimates. 4 Performance Guarantees We provide the performance guarantees for ORL in this section. Throughout this sec- tion, we use following asymptotic inequality notations: for positive numbers a and b, the (cid:46) ≤ asymptotic inequality a b means that a C b where C is a constant only depending p,q p,q p,q on p, q. Suppose N samples, a constant fraction of which are authentic ones and have sub- Gaussian distributions as speciﬁed in (1) for some L, are evenly divided to T mini-batches and outlier fractions on the T mini-batches are λ , . . . , λ respectively. Let θ , . . . , θ be a 1 T 1 T 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algorithm

text: The

label: caption

text: ORL

text: Approach

text: X

text: X

text: Input:

text: sequence

text: convexity

text: parameter

text: c

text: robust

text: learning

text: proce-

text: T

text: a

text: dure

text: parameter

text: ν.

text: Initialization:

text: θ

text: for

text: t

text: T

text: do

text: X

text: Call

text: the

text: robust

text: procedure

text: RL(

text: ν);

text: t

text: Compute

text: weight

text: w

text: 2η

text: θ

text: θ

text: with

text: η

text: t.

text: t

text: t

text: t

text: t

text: a

text: Update

text: the

text: estimate:

text: w

text: )θ

text: w

text: θ

text: t

text: t

text: t

text: t

text: end

text: for

text: Output:

text: Final

text: estimate

text: θ

text: T

text: estimate

text: θ

text: ORL

text: computes

text: the

text: gradient

text: at

text: point

text: θ

text: of

text: the

text: empirical

text: function

text: G

text: in

text: Eqn.

text: t

text: T

text: evaluated

text: only

text: at

text: θ

text: ∂G

text: (θ;

text: θ

text: θ

text: g(θ;

text: θ

text: T

text: ∂θ

text: θ

text: θ

text: Then

text: ORL

text: the

text: following

text: θ

text: θ

text: η

text: g(θ

text: θ

text: w

text: )θ

text: w

text: θ

text: t

text: t

text: t

text: t

text: t

text: t

text: η

text: is

text: a

text: predeﬁned

text: step

text: size

text: parameter

text: which

text: usually

text: takes

text: the

text: form

text: of

text: t

text: with

text: t

text: a

text: a

text: constant

text: c

text: convexity

text: of

text: the

text: empirical

text: function

text: to

text: optimize.

text: Besides,

text: w

text: t

text: 2η

text: θ

text: θ

text: controls

text: contribution

text: of

text: each

text: new

text: estimate

text: θ

text: conservatively

text: in

text: updating

text: the

text: t

text: t

text: global

text: estimate

text: θ.

text: Details

text: of

text: ORL

text: are

text: provided

text: in

text: Algorithm

text: Another

text: level

text: of

text: ﬁltering

text: is

text: important.

text: Certain

text: mini-batches

text: may

text: contain

text: overwhelming

text: outliers.

text: Therefore,

text: even

text: though

text: a

text: robust

text: learning

text: procedure

text: is

text: employed

text: on

text: each

text: mini-batch,

text: the

text: resulted

text: estimate

text: cannot

text: be

text: guaranteed

text: to

text: be

text: accurate.

text: In

text: fact,

text: a

text: mini-batch

text: containing

text: over

text: outliers

text: would

text: corrupt

text: any

text: robust

text: learning

text: procedure—the

text: resulted

text: estimate

text: can

text: be

text: arbitrarily

text: bad

text: and

text: breakdown

text: the

text: overall

text: online

text: learning.

text: To

text: address

text: this

text: critical

text: issue,

text: ORL

text: performs

text: another

text: level

text: of

text: online

text: learning

text: for

text: updating

text: the

text: “global”

text: estimate

text: with

text: adaptive

text: weights

text: for

text: the

text: new

text: estimate

text: and

text: “ﬁlters

text: out”

text: possibly

text: bad

text: estimates.

text: Performance

text: Guarantees

text: We

text: provide

text: the

text: performance

text: guarantees

text: for

text: ORL

text: in

text: this

text: section.

text: Throughout

text: this

text: sec-

text: tion,

text: we

text: use

text: following

text: asymptotic

text: inequality

text: notations:

text: for

text: positive

text: numbers

text: a

text: and

text: b,

text: the

text: asymptotic

text: inequality

text: a

text: b

text: means

text: that

text: a

text: C

text: b

text: where

text: C

text: is

text: a

text: constant

text: only

text: depending

text: p,q

text: p,q

text: p,q

text: on

text: p,

text: q.

text: Suppose

text: N

text: samples,

text: a

text: constant

text: fraction

text: of

text: which

text: are

text: authentic

text: ones

text: and

text: have

text: sub-

text: Gaussian

text: distributions

text: as

text: speciﬁed

text: in

text: for

text: some

text: L,

text: are

text: evenly

text: divided

text: to

text: T

text: mini-batches

text: and

text: outlier

text: fractions

text: on

text: the

text: T

text: mini-batches

text: are

text: λ

text: λ

text: respectively.

text: Let

text: θ

text: θ

text: be

text: a

text: T

text: T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Neutron interference in the Earth’s gravitational ﬁeld Andrei Galiautdinov1 and Lewis H. Ryder2 1Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA 2School of Physical Sciences, University of Kent, Canterbury, Kent, CT2 7NH, United Kingdom (Dated: May 29, 2020) This work relates to the famous experiments, performed in 1975 and 1979 by Werner et al., mea- suring neutron interference and neutron Sagnac eﬀects in the earth’s gravitational ﬁeld. Employing the method of Stodolsky in its weak ﬁeld approximation, explicit expressions are derived for the two phase shifts, which turn out to be in agreement with the experiments and with the previously obtained expressions derived from semi-classical arguments: these expressions are simply modiﬁed by relativistic correction factors. I. INTRODUCTION trons. In Section the general relativistic setting for more realistic derivations of these eﬀects is presented. The Kerr metric is displayed as well as a coordinate transfor- It is now several decades since the ground-breaking mation to a Cartesian system relevant to our problem. In work by Werner and his co-workers showed that gravita- the ﬁnal Section our results are derived. Use is made of tional [1, 2] and rotational [3] eﬀects were to be found the weak ﬁeld approximation in conjunction with a spe- in neutron interference experiments performed on the ciﬁc assumption which allows the calculations to be pre- earth’s surface [4–6]. The predicted and experimentally formed. It is found that the resulting phase shifts are, in conﬁrmed gravitational phase shift is the only expression both cases, those predicted by the simple models in Sec- in physics to feature both Newton’s constant of gravita- − − (cid:126) tions and , with correction factors of γ = (1 v2/c2) 1/2, tion G and Planck’s quantum of action , which surely and additional small terms involving ω, the angular ve- makes these experiments particularly noteworthy. The locity of the earth. two experiments are referred to hereafter as the COW II. SIMPLE DERIVATION OF COW EFFECT experiment and the neutron Sagnac eﬀect. Straightforward, semi-classical derivations of these ef- fects have already appeared in the literature (see for ex- The setup described in reference [2] (see also [9]) is ample [7–9]) and in abbreviated form are summarized in based on the splitting of the neutron beam by Bragg Sections and below. What is very clear, however, is that diﬀraction from perfect crystals, as ﬁrst implemented for a proper account of this topic should really be sought in X rays by Bonse and Hart [18]. Rauch and Werner [9] General Relativity (GR) — which is, after all, a theory of point out that when the desired degree of crystal cut- gravity! — and indeed numerous papers have been writ- ting is achieved, the resulting interferometry ”exhibits ten using this approach (see for example [10–14]). Some the fundamentals of quantum mechanics in a very direct of these explore rather sophisticated notions, for exam- and obvious way”. The interference involved is ”topo- ple a possible parallel between the COW experiment and logically equivalent to a ring”, which we represent as a the Aharonov-Bohm eﬀect, based on the integrated cur- rectangle, of macroscopic dimensions (centimetres). The vature of an enclosed path, on the one hand in parameter neutrons enter at the bottom left corner where the beam space and on the other hand in ﬁeld space [13]. We do splits into two, and the beams recombine at the top right not aim to explore these higher-ﬂown topics, but rather corner, where the interference takes place. to present a simple demonstration of how GR can ac- The spatial part of a plan· e matter wave describing a count for the ﬁndings in neutron interferometry, and, at neutro≡n b| e|am is given by eik r, where k is the wave vector an introductory level suitable for instance for inclusion and k k = 2π/λ is the wave number, with λ being the in an introductory course, demonstrate that general rel- de Broglie wavelength, so the phase accumulated over a ativity has an application in quantum physics [15] — a path from r0 to r is (cid:90) notion which might still cause some surprise! We use the Kerr solution of GR [16, 17], since this r · Φ(r) = k dr, (1) includes the rotation of the earth through the angular momentum parameter a, as well as ω, the angular veloc- r0 ity of the earth, and rs its Schwarzschild radius. These or, since λ = h/p, where p(cid:90)is particle’s momentum, are all small parameters, and we calculate the relevant eﬀects to second order in all these quantities (mixed and unmixed). The general method of procedure is the weak 1 r · Φ(r) = (cid:126) p dr. (2) ﬁeld approximation, adopted by Stodolsky [10]. r0 The next Section describes a standard, elementary derivation of the COW eﬀect, and in Section is a simi- This refers to a particular path, so the phase diﬀerence larly elementary derivation of the Sagnac eﬀect for neu- between neutron beams along two distinct paths is ##LTLine##

text: Neutron

label: title

text: interference

text: in

text: the

text: Earth’s

text: gravitational

text: ﬁeld

text: Andrei

label: author

text: and

text: Lewis

text: H.

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Georgia,

text: Athens,

text: GA

text: USA

text: of

text: Physical

text: Sciences,

text: University

text: of

text: Kent,

text: Canterbury,

text: Kent,

text: United

text: Kingdom

text: (Dated:

text: May

label: date

text: This

label: abstract

text: work

text: relates

text: to

text: the

text: famous

text: experiments,

text: performed

text: in

text: and

text: by

text: Werner

text: et

text: al.,

text: mea-

text: suring

text: neutron

text: interference

text: and

text: neutron

text: Sagnac

text: eﬀects

text: in

text: the

text: earth’s

text: gravitational

text: ﬁeld.

text: Employing

text: the

text: method

text: of

text: Stodolsky

text: in

text: its

text: weak

text: ﬁeld

text: approximation,

text: explicit

text: expressions

text: are

text: derived

text: for

text: the

text: two

text: phase

text: shifts,

text: which

text: turn

text: out

text: to

text: be

text: in

text: agreement

text: with

text: the

text: experiments

text: and

text: with

text: the

text: previously

text: obtained

text: expressions

text: derived

text: from

text: semi-classical

text: arguments:

text: these

text: expressions

text: are

text: simply

text: modiﬁed

text: by

text: relativistic

text: correction

text: factors.

text: I.

text: INTRODUCTION

text: trons.

text: In

text: Section

text: the

text: general

text: relativistic

text: setting

text: for

text: more

text: realistic

text: derivations

text: of

text: these

text: eﬀects

text: is

text: presented.

text: The

text: Kerr

text: metric

text: is

text: displayed

text: as

text: well

text: as

text: a

text: coordinate

text: transfor-

text: It

text: is

text: now

text: several

text: decades

text: since

text: the

text: ground-breaking

text: mation

text: to

text: a

text: Cartesian

text: system

text: relevant

text: to

text: our

text: problem.

text: In

text: work

text: by

text: Werner

text: and

text: his

text: co-workers

text: showed

text: that

text: gravita-

text: the

text: ﬁnal

text: Section

text: our

text: results

text: are

text: derived.

text: Use

text: is

text: made

text: of

text: tional

text: and

text: rotational

text: eﬀects

text: were

text: to

text: be

text: found

text: the

text: weak

text: ﬁeld

text: approximation

text: in

text: conjunction

text: with

text: a

text: spe-

text: in

text: neutron

text: interference

text: experiments

text: performed

text: on

text: the

text: ciﬁc

text: assumption

text: which

text: allows

text: the

text: calculations

text: to

text: be

text: pre-

text: earth’s

text: surface

text: The

text: predicted

text: and

text: experimentally

text: formed.

text: It

text: is

text: found

text: that

text: the

text: resulting

text: phase

text: shifts

text: are,

text: in

text: conﬁrmed

text: gravitational

text: phase

text: shift

text: is

text: the

text: only

text: expression

text: both

text: cases,

text: those

text: predicted

text: by

text: the

text: simple

text: models

text: in

text: Sec-

text: in

text: physics

text: to

text: feature

text: both

text: Newton’s

text: constant

text: of

text: gravita-

text: tions

text: and

text: with

text: correction

text: factors

text: of

text: γ

text: tion

text: G

text: and

text: Planck’s

text: quantum

text: of

text: action

text: which

text: surely

text: and

text: additional

text: small

text: terms

text: involving

text: ω,

text: the

text: angular

text: ve-

text: makes

text: these

text: experiments

text: particularly

text: noteworthy.

text: The

text: locity

text: of

text: the

text: earth.

text: two

text: experiments

text: are

text: referred

text: to

text: hereafter

text: as

text: the

text: COW

text: II.

text: SIMPLE

text: DERIVATION

text: OF

text: COW

text: EFFECT

text: experiment

text: and

text: the

text: neutron

text: Sagnac

text: eﬀect.

text: Straightforward,

text: semi-classical

text: derivations

text: of

text: these

text: ef-

text: fects

text: have

text: already

text: appeared

text: in

text: the

text: literature

text: (see

text: for

text: ex-

text: The

text: setup

text: described

text: in

text: reference

text: (see

text: also

text: is

text: ample

text: and

text: in

text: abbreviated

text: form

text: are

text: summarized

text: in

text: based

text: on

text: the

text: splitting

text: of

text: the

text: neutron

text: beam

text: by

text: Bragg

text: Sections

text: and

text: below.

text: What

text: is

text: very

text: clear,

text: however,

text: is

text: that

text: diﬀraction

text: from

text: perfect

text: crystals,

text: as

text: ﬁrst

text: implemented

text: for

text: a

text: proper

text: account

text: of

text: this

text: topic

text: should

text: really

text: be

text: sought

text: in

text: X

text: rays

text: by

text: Bonse

text: and

text: Hart

text: Rauch

text: and

text: Werner

text: General

text: Relativity

text: (GR)

text: which

text: is,

text: after

text: all,

text: a

text: theory

text: of

text: point

text: out

text: that

text: when

text: the

text: desired

text: degree

text: of

text: crystal

text: cut-

text: gravity!

text: and

text: indeed

text: numerous

text: papers

text: have

text: been

text: writ-

text: ting

text: is

text: achieved,

text: the

text: resulting

text: interferometry

text: ”exhibits

text: ten

text: using

text: this

text: approach

text: (see

text: for

text: example

text: Some

text: the

text: fundamentals

text: of

text: quantum

text: mechanics

text: in

text: a

text: very

text: direct

text: of

text: these

text: explore

text: rather

text: sophisticated

text: notions,

text: for

text: exam-

text: and

text: obvious

text: way”.

text: The

text: interference

text: involved

text: is

text: ”topo-

text: ple

text: a

text: possible

text: parallel

text: between

text: the

text: COW

text: experiment

text: and

text: logically

text: equivalent

text: to

text: a

text: ring”,

text: which

text: we

text: represent

text: as

text: a

text: the

text: Aharonov-Bohm

text: eﬀect,

text: based

text: on

text: the

text: integrated

text: cur-

text: rectangle,

text: of

text: macroscopic

text: dimensions

text: (centimetres).

text: The

text: vature

text: of

text: an

text: enclosed

text: path,

text: on

text: the

text: one

text: hand

text: in

text: parameter

text: neutrons

text: enter

text: at

text: the

text: bottom

text: left

text: corner

text: where

text: the

text: beam

text: space

text: and

text: on

text: the

text: other

text: hand

text: in

text: ﬁeld

text: space

text: We

text: do

text: splits

text: into

text: two,

text: and

text: the

text: beams

text: recombine

text: at

text: the

text: top

text: right

text: not

text: aim

text: to

text: explore

text: these

text: higher-ﬂown

text: topics,

text: but

text: rather

text: corner,

text: where

text: the

text: interference

text: takes

text: place.

text: to

text: present

text: a

text: simple

text: demonstration

text: of

text: how

text: GR

text: can

text: ac-

text: The

text: spatial

text: part

text: of

text: a

text: plan·

text: e

text: matter

text: wave

text: describing

text: a

text: count

text: for

text: the

text: ﬁndings

text: in

text: neutron

text: interferometry,

text: and,

text: at

text: neutro≡n

text: b|

text: e|am

text: is

text: given

text: by

text: eik

text: r,

text: where

text: k

text: is

text: the

text: wave

text: vector

text: an

text: introductory

text: level

text: suitable

text: for

text: instance

text: for

text: inclusion

text: and

text: k

text: k

text: 2π/λ

text: is

text: the

text: wave

text: number,

text: with

text: λ

text: being

text: the

text: in

text: an

text: introductory

text: course,

text: demonstrate

text: that

text: general

text: rel-

text: de

text: Broglie

text: wavelength,

text: so

text: the

text: phase

text: accumulated

text: over

text: a

text: ativity

text: has

text: an

text: application

text: in

text: quantum

text: physics

text: a

text: path

text: from

text: to

text: r

text: is

text: notion

text: which

text: might

text: still

text: cause

text: some

text: surprise!

text: We

text: use

text: the

text: Kerr

text: solution

text: of

text: GR

text: since

text: this

text: r

text: Φ(r)

text: k

text: dr,

text: includes

text: the

text: rotation

text: of

text: the

text: earth

text: through

text: the

text: angular

text: momentum

text: parameter

text: a,

text: as

text: well

text: as

text: ω,

text: the

text: angular

text: veloc-

text: ity

text: of

text: the

text: earth,

text: and

text: rs

text: its

text: Schwarzschild

text: radius.

text: These

text: or,

text: since

text: λ

text: h/p,

text: where

text: particle’s

text: momentum,

text: are

text: all

text: small

text: parameters,

text: and

text: we

text: calculate

text: the

text: relevant

text: eﬀects

text: to

text: second

text: order

text: in

text: all

text: these

text: quantities

text: (mixed

text: and

text: unmixed).

text: The

text: general

text: method

text: of

text: procedure

text: is

text: the

text: weak

text: r

text: Φ(r)

text: p

text: dr.

text: ﬁeld

text: approximation,

text: adopted

text: by

text: Stodolsky

text: The

text: next

text: Section

text: describes

text: a

text: standard,

text: elementary

text: derivation

text: of

text: the

text: COW

text: eﬀect,

text: and

text: in

text: Section

text: is

text: a

text: simi-

text: This

text: refers

text: to

text: a

text: particular

text: path,

text: so

text: the

text: phase

text: diﬀerence

text: larly

text: elementary

text: derivation

text: of

text: the

text: Sagnac

text: eﬀect

text: for

text: neu-

text: between

text: neutron

text: beams

text: along

text: two

text: distinct

text: paths

text: is

text: ##LTLine##


text: ⇒ Proof. (i) (ii) Let t be the term given by the loop condition. By compatibility, if we apply t to a tuple of edges, the result is also an edge. We will use the edges of G which are images of edges of GC in the order from the construction of GC. By the loop condition the initial vertex is equal to the terminal vertex of the resulting edge so there is a loop in G. ⇒ (ii) (iii) Trivial. ⇒ V (iii) (i) Let F be the -free algebra generated by vertices of GC. Let E be the subuniverse of F2 generated by edges of GC. Set E forms edges of a graph compatible with F containing GC as a subgraph. So by (iii) there is loop (a, a) in E. The pair (a, a) is generated by generators of E, so there is a term operation t taking edges of GC as parameters and returning (a, a). Thus in F, the term t satisﬁes the loop condition if we plug generators of F to it. So by C universality of free algebra, the term t satisﬁes in general. C D Corollary 1. Assume loop conditions , . If there is a graph homomorphism → C D C D C D GC GD, then implies . Namely implies if is a subgraph of . Finally, we recall a standard method for building compatible relations (graphs) from existing ones – pp-deﬁnitions and pp-powers. A relation R is pp-deﬁnable from relations R , . . . , R if it can be described by a formula using variables, 1 n existential quantiﬁers, conjunctions and predicates R , . . . , R . If R , . . . , R 1 n 1 n are compatible with an algebra, then also R is. Using graph view, pp-deﬁnition of a k-ary relation R from graphs G , . . . , G 1 n sharing one set of vertices V is given by a graph on a set U of vertices with n types of edges and with k distinguished vertices u , . . . , u . Then vertices ∈ → 1 k v , . . . , v V are R-related if there is a mapping U V which sends u , . . . , u 1 k 1 k to v , . . . , v respectively and which maps edges of i-th type to edges in G . 1 k · i Now assume a pp-deﬁned k l-ary relation on A. We can understand it as a k-ary relation on the set Al. In this case, we call it pp-power. If original relations are compatible with an algebra A, then the pp-power is compatible with the algebraic power Al. 4 Unoriented case In this section, we will focus on loop conditions with unoriented graph. By proposition 1, loop equations with isomorphic graphs conditions are equivalent. Therefore, we will describe loop conditions by its graphs. Namely, we have loop conditions: edge (commutativity), triangle (Siggers), more gener- ally n-clique and n-cycles. At the begining, we can use just corollary 1 to get some implications be- tween unoriented loop conditions. By deﬁnition, a loop condition cannot be empty, so every unoriented loop condition contains an unoriented edge. Thus edge is the strongest unoriented loop condition. Every bipartite graph can be homomorphically mapped to an edge, so all bipartite graphs loop conditions are equivalent. Triangle implies 4-clique, it implies 5-clique and so on. Moreover every loopless graph is a subgraph of a clique large enough, so the chain of cliques reaches the weakest non-trivial loop conditions (not only non-oriented). 5-cycle can be mapped to triangle, 7-cycle can be mapped to 5 cycle and so on. Moreover every non-bipartite graph contains an odd cycle as a subgraph. 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: (i)

text: (ii)

text: Let

text: t

text: be

text: the

text: term

text: given

text: by

text: the

text: loop

text: condition.

text: By

text: compatibility,

text: if

text: we

text: apply

text: t

text: to

text: a

text: tuple

text: of

text: edges,

text: the

text: result

text: is

text: also

text: an

text: edge.

text: We

text: will

text: use

text: the

text: edges

text: of

text: G

text: which

text: are

text: images

text: of

text: edges

text: of

text: GC

text: in

text: the

text: order

text: from

text: the

text: construction

text: of

text: GC.

text: By

text: the

text: loop

text: condition

text: the

text: initial

text: vertex

text: is

text: equal

text: to

text: the

text: terminal

text: vertex

text: of

text: the

text: resulting

text: edge

text: so

text: there

text: is

text: a

text: loop

text: in

text: G.

text: (ii)

text: (iii)

text: Trivial.

text: V

text: (iii)

text: (i)

text: Let

text: F

text: be

text: the

text: -free

text: algebra

text: generated

text: by

text: vertices

text: of

text: GC.

text: Let

text: E

text: be

text: the

text: subuniverse

text: of

text: generated

text: by

text: edges

text: of

text: GC.

text: Set

text: E

text: forms

text: edges

text: of

text: a

text: graph

text: compatible

text: with

text: F

text: containing

text: GC

text: as

text: a

text: subgraph.

text: So

text: by

text: (iii)

text: there

text: is

text: loop

text: (a,

text: a)

text: in

text: E.

text: The

text: pair

text: (a,

text: a)

text: is

text: generated

text: by

text: generators

text: of

text: E,

text: so

text: there

text: is

text: a

text: term

text: operation

text: t

text: taking

text: edges

text: of

text: GC

text: as

text: parameters

text: and

text: returning

text: (a,

text: a).

text: Thus

text: in

text: F,

text: the

text: term

text: t

text: satisﬁes

text: the

text: loop

text: condition

text: if

text: we

text: plug

text: generators

text: of

text: F

text: to

text: it.

text: So

text: by

text: C

text: universality

text: of

text: free

text: algebra,

text: the

text: term

text: t

text: satisﬁes

text: in

text: general.

text: C

text: D

text: Corollary

text: Assume

text: loop

text: conditions

text: If

text: there

text: is

text: a

text: graph

text: homomorphism

text: C

text: D

text: C

text: D

text: C

text: D

text: GC

text: GD,

text: then

text: implies

text: Namely

text: implies

text: if

text: is

text: a

text: subgraph

text: of

text: Finally,

text: we

text: recall

text: a

text: standard

text: method

text: for

text: building

text: compatible

text: relations

text: (graphs)

text: from

text: existing

text: ones

text: pp-deﬁnitions

text: and

text: pp-powers.

text: A

text: relation

text: R

text: is

text: pp-deﬁnable

text: from

text: relations

text: R

text: R

text: if

text: it

text: can

text: be

text: described

text: by

text: a

text: formula

text: using

text: variables,

text: n

text: existential

text: quantiﬁers,

text: conjunctions

text: and

text: predicates

text: R

text: R

text: If

text: R

text: R

text: n

text: n

text: are

text: compatible

text: with

text: an

text: algebra,

text: then

text: also

text: R

text: is.

text: Using

text: graph

text: view,

text: pp-deﬁnition

text: of

text: a

text: k-ary

text: relation

text: R

text: from

text: graphs

text: G

text: G

text: n

text: sharing

text: one

text: set

text: of

text: vertices

text: V

text: is

text: given

text: by

text: a

text: graph

text: on

text: a

text: set

text: U

text: of

text: vertices

text: with

text: n

text: types

text: of

text: edges

text: and

text: with

text: k

text: distinguished

text: vertices

text: u

text: u

text: Then

text: vertices

text: k

text: v

text: v

text: V

text: are

text: R-related

text: if

text: there

text: is

text: a

text: mapping

text: U

text: V

text: which

text: sends

text: u

text: u

text: k

text: k

text: to

text: v

text: v

text: respectively

text: and

text: which

text: maps

text: edges

text: of

text: i-th

text: type

text: to

text: edges

text: in

text: G

text: k

text: i

text: Now

text: assume

text: a

text: pp-deﬁned

text: k

text: l-ary

text: relation

text: on

text: A.

text: We

text: can

text: understand

text: it

text: as

text: a

text: k-ary

text: relation

text: on

text: the

text: set

text: Al.

text: In

text: this

text: case,

text: we

text: call

text: it

text: pp-power.

text: If

text: original

text: relations

text: are

text: compatible

text: with

text: an

text: algebra

text: A,

text: then

text: the

text: pp-power

text: is

text: compatible

text: with

text: the

text: algebraic

text: power

text: Al.

text: Unoriented

text: case

text: In

text: this

text: section,

text: we

text: will

text: focus

text: on

text: loop

text: conditions

text: with

text: unoriented

text: graph.

text: By

text: proposition

text: loop

text: equations

text: with

text: isomorphic

text: graphs

text: conditions

text: are

text: equivalent.

text: Therefore,

text: we

text: will

text: describe

text: loop

text: conditions

text: by

text: its

text: graphs.

text: Namely,

text: we

text: have

text: loop

text: conditions:

text: edge

text: (commutativity),

text: triangle

text: (Siggers),

text: more

text: gener-

text: ally

text: n-clique

text: and

text: n-cycles.

text: At

text: the

text: begining,

text: we

text: can

text: use

text: just

text: corollary

text: to

text: get

text: some

text: implications

text: be-

text: tween

text: unoriented

text: loop

text: conditions.

text: By

text: deﬁnition,

text: a

text: loop

text: condition

text: cannot

text: be

text: empty,

text: so

text: every

text: unoriented

text: loop

text: condition

text: contains

text: an

text: unoriented

text: edge.

text: Thus

text: edge

text: is

text: the

text: strongest

text: unoriented

text: loop

text: condition.

text: Every

text: bipartite

text: graph

text: can

text: be

text: homomorphically

text: mapped

text: to

text: an

text: edge,

text: so

text: all

text: bipartite

text: graphs

text: loop

text: conditions

text: are

text: equivalent.

text: Triangle

text: implies

text: it

text: implies

text: and

text: so

text: on.

text: Moreover

text: every

text: loopless

text: graph

text: is

text: a

text: subgraph

text: of

text: a

text: clique

text: large

text: enough,

text: so

text: the

text: chain

text: of

text: cliques

text: reaches

text: the

text: weakest

text: non-trivial

text: loop

text: conditions

text: (not

text: only

text: non-oriented).

text: can

text: be

text: mapped

text: to

text: triangle,

text: can

text: be

text: mapped

text: to

text: cycle

text: and

text: so

text: on.

text: Moreover

text: every

text: non-bipartite

text: graph

text: contains

text: an

text: odd

text: cycle

text: as

text: a

text: subgraph.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: These implications cannot be reversed even for idempotent ﬁnite case. An example of an algebra satisfying (2) but not (1) is the algebra (0, 1, 2, m) where − m(x, y, z) = x + y z. 5 Unoriented case – open An oriented graph G is said to be smooth if every vertex has an incoming and an outcoming edge. G is said to have the algebraic length 1 if there is no graph homomorphism from G to a directed cycle of a length greater than one. The following theorem holds for ﬁnite algebras [1]. Theorem 1. Let A be a ﬁnite algebra. Let G, H be weakly connected smooth graphs with algebraic length one. Then A satisﬁes G loop condition if and only if it satisﬁes H loop conditions. Less formally, all connected smooth loop conditions with algebraic length 1 are equivalent for ﬁnite algebras. It gives a simpler weakest loop condition than triangle for ﬁnite algebras: s(a, r, e, a) = s(r, a, r, e). We have no idea whether the theorem or the corollary can hold in general. We recommend the following question for the beginning. ? ? ⇔ ⇔ Figure 6: Open problems in loop conditions, known for ﬁnite algebras. Open problem 1. Let G be an unoriented triangle and H be a triangle with two unoriented edges and one oriented edge. Does G imply H? References [1] Keith Kearnes, Petar Markovi´c, and Ralph McKenzie. Optimal strong Mal’cev conditions for omitting type 1 in locally ﬁnite varieties. Algebra Universalis, 72(1):91–100, 2014. [2] Olˇs´ak Miroslav. The weakest nontrivial idempotent equations. manuscript, 2016. [3] Mark H. Siggers. A strong Mal’cev condition for locally ﬁnite varieties omit- ting the unary type. Algebra Universalis, 64(1-2):15–20, 2010. 7 ##LTFigure##

text: These

text: implications

text: cannot

text: be

text: reversed

text: even

text: for

text: idempotent

text: ﬁnite

text: case.

text: An

text: example

text: of

text: an

text: algebra

text: satisfying

text: but

text: not

text: is

text: the

text: algebra

text: m)

text: where

text: m(x,

text: y,

text: z)

text: x

text: y

text: z.

text: Unoriented

text: case

text: open

text: An

text: oriented

text: graph

text: G

text: is

text: said

text: to

text: be

text: smooth

text: if

text: every

text: vertex

text: has

text: an

text: incoming

text: and

text: an

text: outcoming

text: edge.

text: G

text: is

text: said

text: to

text: have

text: the

text: algebraic

text: length

text: if

text: there

text: is

text: no

text: graph

text: homomorphism

text: from

text: G

text: to

text: a

text: directed

text: cycle

text: of

text: a

text: length

text: greater

text: than

text: one.

text: The

text: following

text: theorem

text: holds

text: for

text: ﬁnite

text: algebras

text: Theorem

text: Let

text: A

text: be

text: a

text: ﬁnite

text: algebra.

text: Let

text: G,

text: H

text: be

text: weakly

text: connected

text: smooth

text: graphs

text: with

text: algebraic

text: length

text: one.

text: Then

text: A

text: satisﬁes

text: G

text: loop

text: condition

text: if

text: and

text: only

text: if

text: it

text: satisﬁes

text: H

text: loop

text: conditions.

text: Less

text: formally,

text: all

text: connected

text: smooth

text: loop

text: conditions

text: with

text: algebraic

text: length

text: are

text: equivalent

text: for

text: ﬁnite

text: algebras.

text: It

text: gives

text: a

text: simpler

text: weakest

text: loop

text: condition

text: than

text: triangle

text: for

text: ﬁnite

text: algebras:

text: s(a,

text: r,

text: e,

text: a)

text: s(r,

text: a,

text: r,

text: e).

text: We

text: have

text: no

text: idea

text: whether

text: the

text: theorem

text: or

text: the

text: corollary

text: can

text: hold

text: in

text: general.

text: We

text: recommend

text: the

text: following

text: question

text: for

text: the

text: beginning.

text: Figure

text: Open

text: problems

text: in

text: loop

text: conditions,

text: known

text: for

text: ﬁnite

text: algebras.

text: Open

text: problem

text: Let

text: G

text: be

text: an

text: unoriented

text: triangle

text: and

text: H

text: be

text: a

text: triangle

text: with

text: two

text: unoriented

text: edges

text: and

text: one

text: oriented

text: edge.

text: Does

text: G

text: imply

text: H?

text: References

text: Keith

text: Kearnes,

text: Petar

text: Markovi´c,

text: and

text: Ralph

text: McKenzie.

text: Optimal

text: strong

text: Mal’cev

text: conditions

text: for

text: omitting

text: type

text: in

text: locally

text: ﬁnite

text: varieties.

text: Algebra

text: Universalis,

text: Olˇs´ak

text: Miroslav.

text: The

text: weakest

text: nontrivial

text: idempotent

text: equations.

text: manuscript,

text: Mark

text: H.

text: Siggers.

text: A

text: strong

text: Mal’cev

text: condition

text: for

text: locally

text: ﬁnite

text: varieties

text: omit-

text: ting

text: the

text: unary

text: type.

text: Algebra

text: Universalis,

text: ##LTFigure##