text: M A SIMPLICIAL APPROACH TO EFFECTIVE DIVISORS IN 0,N 17 Figure 4. A cycle of tetrehedra. Figure 5. Triangulated surface of an octahedron (image source: Wikipedia.org). {± } weights in 1 . Minimality is clear since, as usual, removing triangles yields edges contained in a single triangle. Since this non-singular complex has 18 vertices, we therefore obtain by Theorem M ≥ 3.25 an irreducible divisor class, hence Cox ring generator, over any ﬁeld, for with n 19. 0,n Indeed, since this 2-complex is not the suspension of a graph it does not decompose into a product of disjointly supported non-singular minimal complexes. By triangulating the six faces of a cube into four triangles each, as depicted in Figure 6, we obtain {± } a minimal 2-complex that can be balanced with weights in 1 and which is not the product of minimal complexes. The analysis is quite similar to the triangulated octahedron described above. M ≥ Here we obtain an irreducible divisor class on for all n 15. 0,n The truncated octahedron has 8 hexagonal faces and 6 square faces (see Figure 7). By subdividing each square face into 4 triangles as before, and each hexagonal face into 6 triangles meeting at a central vertex, we again obtain a minimal complex with irreducible divisor class, now deﬁned on M ≥ for n 39. 0,n The sphere does not seem to be special with regard to triangulations yielding minimal complexes. For instance, we can triangulate the torus as shown in Figure 8 to obtain an irreducible divisor M ≥ class on whenever n 10. 0,n ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: M A SIMPLICIAL APPROACH TO EFFECTIVE DIVISORS IN 0,N 3 M M − perspective on the Cox ring Cox( ) and eﬀective monoid M( ), and, like the case of ( 1)- 0,n 0,n M curves on a del Pezzo surface, it ties the geometry of eﬀective divisors in to many beautiful 0,n classical constructions, such as triangulations of the sphere and other closed manifolds. We list § several open questions and areas of further study arising from this perspective in . ∼ M P − Throughout this paper we ﬁx an isomorphism = Bl n 3, i.e., a choice of ψ-class. This 0,n C isomorphism was originally provided by Kapranov over Spec [Kap93a], but work of Hassett Z § Z M extends it to be deﬁned over Spec [Has03, 6]. This yields a -grading on Cox( ) and M 0,n Pic(M0,n);→indeedP, b−y th∼e dZegree of a divisor or its class we shall mean the image under the map Pic( ) Pic( n 3) = induced by the above blow-up presentation. Thus, the degree of an 0,n eﬀective divisor is zero if it is a union of exceptional divisors and it is simply the degree of its image P − as a hypersurface in n 3 otherwise. M The ﬁrst conjecture concerning eﬀective divisors in was given by Fulton, although formally 0,n M stated by Keel and McKernan in [KM96]. It asserts that Eﬀ( ) is spanned by the classes of 0,n boundary divisors, extending the del Pezzo situation of n = 5 and inspired by the geometry of toric varieties where boundary strata do indeed span all eﬀective classes. Put another way, this conjecture says that the eﬀective cone is generated by classes of degree at most one. However, Keel and Vermeire simultaneously found degree two counterexamples for n = 6 [Ver02, GKM02]. Hassett and Tschinkel, with the help of computer calculation, veriﬁed that for n = 6 the Keel-Vermeire classes, together with the boundary classes, do in fact span the entire eﬀective cone [HT02]. The question remained of whether the unique (up to scalar) sections of these divisor classes M generate Cox( ). Here Castravet performed a comprehensive analysis of this particular case and 0,6 M found a positive answer to this question [Cas09]. This implies that the eﬀective monoid M( ) 0,6 is also minimally generated by the boundary and Keel-Vermeire classes, so essentially everything is known here and attention now turns to n > 6. Many experts believed the Keel-Vermeire classes in n = 6 must be part of a more general construction valid for all n, yet this vast generalization proved rather elusive. A breakthrough was made in 2009 when Castravet and Tevelev found a § construction (summarized in ) that yields all Keel-Vermeire classes in n = 6 and a huge number ≥ of new extremal rays for n 7. They conjectured that the boundary classes together with these M so-called hypertree divisor classes span Eﬀ( ) for all n [CT13a]. Additionally, they raised the 0,n question (and called it a “pipe dream”) of whether, as with n = 6, it is always the case that the Cox ring is generated by sections of divisor classes spanning extremal rays of the eﬀective cone. M One could weaken this dream slightly and ask for the eﬀective monoid M( ) to be generated 0,n ≥ by classes lying along extremal rays. As we shall see below, these dreams fail for all n 7, as does the hypertree conjecture (see also Remark 1.2 concerning Opie’s exciting paper [Opi13]). M Along with the remarkable recent result of Castravet and Tevelev that Cox( ) is not ﬁnitely ≥ 0M,n generated for n 134 [CT13b], or in other words, that Hu and Keel’s dream for is also false 0,n [HK00], we are left once again in a quandary: one hopes that there is some order to the seemingly M chaotic and unwieldy collection of eﬀective divisors and divisor classes on , yet the challenge 0,n is to ﬁnd a structure that best describes it and reveals how truly complex it is. For us, a (possibly singular) d-simplex is a multiset of cardinality d + 1, and a pure-dimensional − { − } simplicial complex of dimension d, or “d-complex”, with vertex set [n 1] := 1, . . . , n 1 is any − set of d-simplices with entries in [n 1]. Thus, a 1-complex is a graph with loops allowed but ∈ multiple edges disallowed. For a multiset S, the number of times i S occurs is its multiplicity, denoted mult (S). We call a complex “non-singular” if every element of every simplex in it has i multiplicity at most one. M Fix a ring R and consider relative to Spec R. A weighted complex is an assignment of a 0,n nonzero element of R to each simplex in a complex, and a weighted complex is “balanced” if, for ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: M

label: paragraph

text: A

text: SIMPLICIAL

text: APPROACH

text: TO

text: EFFECTIVE

text: DIVISORS

text: IN

text: M

text: M

text: perspective

text: on

text: the

text: Cox

text: ring

text: Cox(

text: and

text: eﬀective

text: monoid

text: M(

text: and,

text: like

text: the

text: case

text: of

text: M

text: curves

text: on

text: a

text: del

text: Pezzo

text: surface,

text: it

text: ties

text: the

text: geometry

text: of

text: eﬀective

text: divisors

text: in

text: to

text: many

text: beautiful

text: classical

text: constructions,

text: such

text: as

text: triangulations

text: of

text: the

text: sphere

text: and

text: other

text: closed

text: manifolds.

text: We

text: list

text: several

text: open

text: questions

text: and

text: areas

text: of

text: further

text: study

text: arising

text: from

text: this

text: perspective

text: in

text: M

text: P

text: Throughout

text: this

text: paper

text: we

text: ﬁx

text: an

text: isomorphism

text: Bl

text: n

text: i.e.,

text: a

text: choice

text: of

text: ψ-class.

text: This

text: C

text: isomorphism

text: was

text: originally

text: provided

text: by

text: Kapranov

text: over

text: Spec

text: but

text: work

text: of

text: Hassett

text: Z

text: Z

text: M

text: extends

text: it

text: to

text: be

text: deﬁned

text: over

text: Spec

text: This

text: yields

text: a

text: -grading

text: on

text: Cox(

text: and

text: M

text: b−y

text: th∼e

text: dZegree

text: of

text: a

text: divisor

text: or

text: its

text: class

text: we

text: shall

text: mean

text: the

text: image

text: under

text: the

text: map

text: Pic(

text: Pic(

text: n

text: induced

text: by

text: the

text: above

text: blow-up

text: presentation.

text: Thus,

text: the

text: degree

text: of

text: an

text: eﬀective

text: divisor

text: is

text: zero

text: if

text: it

text: is

text: a

text: union

text: of

text: exceptional

text: divisors

text: and

text: it

text: is

text: simply

text: the

text: degree

text: of

text: its

text: image

text: P

text: as

text: a

text: hypersurface

text: in

text: n

text: otherwise.

text: M

text: The

text: ﬁrst

text: conjecture

text: concerning

text: eﬀective

text: divisors

text: in

text: was

text: given

text: by

text: Fulton,

text: although

text: formally

text: M

text: stated

text: by

text: Keel

text: and

text: McKernan

text: in

text: It

text: asserts

text: that

text: Eﬀ(

text: is

text: spanned

text: by

text: the

text: classes

text: of

text: boundary

text: divisors,

text: extending

text: the

text: del

text: Pezzo

text: situation

text: of

text: n

text: and

text: inspired

text: by

text: the

text: geometry

text: of

text: toric

text: varieties

text: where

text: boundary

text: strata

text: do

text: indeed

text: span

text: all

text: eﬀective

text: classes.

text: Put

text: another

text: way,

text: this

text: conjecture

text: says

text: that

text: the

text: eﬀective

text: cone

text: is

text: generated

text: by

text: classes

text: of

text: degree

text: at

text: most

text: one.

text: However,

text: Keel

text: and

text: Vermeire

text: simultaneously

text: found

text: degree

text: two

text: counterexamples

text: for

text: n

text: Hassett

text: and

text: Tschinkel,

text: with

text: the

text: help

text: of

text: computer

text: calculation,

text: veriﬁed

text: that

text: for

text: n

text: the

text: Keel-Vermeire

text: classes,

text: together

text: with

text: the

text: boundary

text: classes,

text: do

text: in

text: fact

text: span

text: the

text: entire

text: eﬀective

text: cone

text: The

text: question

text: remained

text: of

text: whether

text: the

text: unique

text: (up

text: to

text: scalar)

text: sections

text: of

text: these

text: divisor

text: classes

text: M

text: generate

text: Cox(

text: Here

text: Castravet

text: performed

text: a

text: comprehensive

text: analysis

text: of

text: this

text: particular

text: case

text: and

text: M

text: found

text: a

text: positive

text: answer

text: to

text: this

text: question

text: This

text: implies

text: that

text: the

text: eﬀective

text: monoid

text: M(

text: is

text: also

text: minimally

text: generated

text: by

text: the

text: boundary

text: and

text: Keel-Vermeire

text: classes,

text: so

text: essentially

text: everything

text: is

text: known

text: here

text: and

text: attention

text: now

text: turns

text: to

text: n

text: Many

text: experts

text: believed

text: the

text: Keel-Vermeire

text: classes

text: in

text: n

text: must

text: be

text: part

text: of

text: a

text: more

text: general

text: construction

text: valid

text: for

text: all

text: n,

text: yet

text: this

text: vast

text: generalization

text: proved

text: rather

text: elusive.

text: A

text: breakthrough

text: was

text: made

text: in

text: when

text: Castravet

text: and

text: Tevelev

text: found

text: a

text: construction

text: (summarized

text: in

text: that

text: yields

text: all

text: Keel-Vermeire

text: classes

text: in

text: n

text: and

text: a

text: huge

text: number

text: of

text: new

text: extremal

text: rays

text: for

text: n

text: They

text: conjectured

text: that

text: the

text: boundary

text: classes

text: together

text: with

text: these

text: M

text: so-called

text: hypertree

text: divisor

text: classes

text: span

text: Eﬀ(

text: for

text: all

text: n

text: Additionally,

text: they

text: raised

text: the

text: question

text: (and

text: called

text: it

text: a

text: “pipe

text: dream”)

text: of

text: whether,

text: as

text: with

text: n

text: it

text: is

text: always

text: the

text: case

text: that

text: the

text: Cox

text: ring

text: is

text: generated

text: by

text: sections

text: of

text: divisor

text: classes

text: spanning

text: extremal

text: rays

text: of

text: the

text: eﬀective

text: cone.

text: M

text: One

text: could

text: weaken

text: this

text: dream

text: slightly

text: and

text: ask

text: for

text: the

text: eﬀective

text: monoid

text: M(

text: to

text: be

text: generated

text: by

text: classes

text: lying

text: along

text: extremal

text: rays.

text: As

text: we

text: shall

text: see

text: below,

text: these

text: dreams

text: fail

text: for

text: all

text: n

text: as

text: does

text: the

text: hypertree

text: conjecture

text: (see

text: also

text: Remark

text: concerning

text: Opie’s

text: exciting

text: paper

text: M

text: Along

text: with

text: the

text: remarkable

text: recent

text: result

text: of

text: Castravet

text: and

text: Tevelev

text: that

text: Cox(

text: is

text: not

text: ﬁnitely

text: generated

text: for

text: n

text: or

text: in

text: other

text: words,

text: that

text: Hu

text: and

text: Keel’s

text: dream

text: for

text: is

text: also

text: false

text: we

text: are

text: left

text: once

text: again

text: in

text: a

text: quandary:

text: one

text: hopes

text: that

text: there

text: is

text: some

text: order

text: to

text: the

text: seemingly

text: M

text: chaotic

text: and

text: unwieldy

text: collection

text: of

text: eﬀective

text: divisors

text: and

text: divisor

text: classes

text: on

text: yet

text: the

text: challenge

text: is

text: to

text: ﬁnd

text: a

text: structure

text: that

text: best

text: describes

text: it

text: and

text: reveals

text: how

text: truly

text: complex

text: it

text: is.

text: For

text: us,

text: a

text: (possibly

text: singular)

text: d-simplex

text: is

text: a

text: multiset

text: of

text: cardinality

text: d

text: and

text: a

text: pure-dimensional

text: simplicial

text: complex

text: of

text: dimension

text: d,

text: or

text: “d-complex”,

text: with

text: vertex

text: set

text: [n

text: n

text: is

text: any

text: set

text: of

text: d-simplices

text: with

text: entries

text: in

text: [n

text: Thus,

text: a

text: is

text: a

text: graph

text: with

text: loops

text: allowed

text: but

text: multiple

text: edges

text: disallowed.

text: For

text: a

text: multiset

text: S,

text: the

text: number

text: of

text: times

text: i

text: S

text: occurs

text: is

text: its

text: multiplicity,

text: denoted

text: mult

text: (S).

text: We

text: call

text: a

text: complex

text: “non-singular”

text: if

text: every

text: element

text: of

text: every

text: simplex

text: in

text: it

text: has

text: i

text: multiplicity

text: at

text: most

text: one.

text: M

text: Fix

text: a

text: ring

text: R

text: and

text: consider

text: relative

text: to

text: Spec

text: R.

text: A

text: weighted

text: complex

text: is

text: an

text: assignment

text: of

text: a

text: nonzero

text: element

text: of

text: R

text: to

text: each

text: simplex

text: in

text: a

text: complex,

text: and

text: a

text: weighted

text: complex

text: is

text: “balanced”

text: if,

text: for

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text: Then, by evaluating this equation at B so that we can esti- Basis points learned Basis points estimated by mate the values of φj(b ) and λ , we obtain the j-th eigen- from K-means evidence maximization i j function φj(x) as follows 2 2 1 1 √ ns φj(x) = λ(MM) k(x)u(jM) = k(x)uj (5) Predictioy−−210 y−−210 j −3 −3 (cid:44) 0 2 4 6 0 2 4 6 where k(x) [k(x, b ), . . . , k(x, b )], λ(M) and u(M) 1 x 1 x 1 M j j are the j-th eigenvalue and eige√nvector of the covariance func- ns 0.5 0.5 o tion evaluated at B, and uj = Mu(jM)/λ(jM). Note that, for ncti 0 0 simplicity, we have chosen the number of the inducing inputs nfuy−0.5 y−0.5 e to be the same as the number of the eigenfunctions; in prac- g Ei −1 −1 tice, we can use more inducing inputs while computing only −1.5 −1.5 the top M eigenvectors. As shown in (5), our eigenfunction 0 2 x 4 6 0 2 x 4 6 lives in a RKHS as a linear combination of the kernel func- tions evaluated at B with weights u . j Inserting (5) into (1) we obtain Figure 1: Illustration of the effect of optimizing basis points (cid:88) (cid:88) (i.e., inducing inputs). In the ﬁrst row, the pink dots represent data points, the blue and red solid curves correspond to the M M ± predictive mean of EigenGP and two standard deviations f(x) = α u k(x, b ) (6) j ij i around the mean, the black curve corresponds to the predic- j=1 i=1 tive mean of the full GP, and the black crosses denote the basis points. In the second row, the curves in various colors repre- This equation reveals a two-layer structure of EigenGP. sent the ﬁve eigenfunctions of EigenGP. The ﬁrst layer linearly combines multiple kernel functions to generate each eigenfunction φj. The second layer takes these eigenfunctions as the basis functions to generate the function { } ARD kernel). Thus, this model can accurately model the un- value f. Note that f is a Bayesian linear combination of φj certainty of a test point even when it is far from the training where the weights α are integrated out to avoid overﬁtting. set. We derive the optimization updates of all the hyperparam- All the model hyperparameters are learned from data. eters for both the original and modiﬁed EigenGP models. But Speciﬁcally, for the ﬁrst layer, to learn the eigenfunctions { } according to our experiments, the modiﬁed model does not φ , we estimate the inducing inputs B and the kernel hy- i improve the prediction accuracy over the original EigenGP (it perparameters (such as lengthscales η for the ARD kernel) by even reduces the accuracy sometimes.). Therefore, we will maximizing the model marginal likelihood. For the second focus on the original EigenGP model in our presentation for layer, we marginalize out α to avoid overﬁtting and maximize its simplicity. Before we present details about hyperparame- the model marginal likelihood to learn the hyperparameter w ter optimization, let us ﬁrst look at an illustrative example on of the prior. the effect of hyperparameter optimization, in particular, the With the estimated hyperparameters, the prior over f is optimization of the inducing inputs B. nonstationary because its covariance function in (2) varies at 3.1 Illustrative Example different regions of x. This comes at no surprise since the eigenfunctions are tied with p(x) in (3). This nonstationarity For this example, we consider a toy dataset used by the reﬂects the fact that our model is adaptive to the distribution FITC algorithm [Snelson and Ghahramani, 2006]. It contains of the explanatory variables x. 200 one-dimensional training samples. We use 5 basis points Note that to recover the full uncertainty captured by the ker- (M = 5) and choose the ARD kernel. We then compare the nel function k, we can add the following term into the kernel { } basis functions φj and the corresponding predictive distri- function of EigenGP: butions in two cases. For the ﬁrst case, we use the kernel width η learned from the full GP model as the kernel width for δ(x − x(cid:48))(k(x, x(cid:48)) − k˜(x, x(cid:48))) (7) EigenGP, and apply K-means to set the basis points B as clus- ter centers. The idea of using K-means to set the basis points has been suggested by Zhang et al. [2008] to minimize an where δ(a) = 1 if and only if a = 0. Compared to the orig- error bound for the Nystro¨m approximation. For the second inal EigenGP model, which has a ﬁnite degree of freedom, case, we optimize η, B and w by maximizing the marginal this modiﬁed model has the inﬁnite number of basis functions likelihood of EigenGP. (assuming k has an inﬁnite number of basis functions as the The results are shown in Figure 1. The ﬁrst row demon- 3 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Then,

text: by

text: evaluating

text: this

text: equation

text: at

text: B

text: so

text: that

text: we

text: can

text: esti-

text: Basis

font: default

text: points

text: learned

text: Basis

text: points

text: estimated

text: by

text: mate

text: the

text: values

text: of

text: φj(b

text: and

text: λ

text: we

text: obtain

text: the

text: j-th

text: eigen-

text: from

text: K-means

text: evidence

text: maximization

text: i

text: j

text: function

text: φj(x)

text: as

text: follows

label: equation

text: ns

text: φj(x)

text: λ(MM)

text: k(x)u(jM)

text: k(x)uj

text: j

text: −3

text: −3

text: where

text: k(x)

text: [k(x,

text: b

text: k(x,

text: b

text: λ(M)

text: and

text: u(M)

text: x

text: x

text: M

text: j

text: j

text: are

text: the

text: j-th

text: eigenvalue

text: and

text: eige√nvector

text: of

text: the

text: covariance

text: func-

text: ns

text: o

text: tion

text: evaluated

text: at

text: B,

text: and

text: uj

text: Mu(jM)/λ(jM).

text: Note

text: that,

text: for

text: ncti

text: simplicity,

text: we

text: have

text: chosen

text: the

text: number

text: of

text: the

text: inducing

text: inputs

text: e

text: to

text: be

text: the

text: same

text: as

text: the

text: number

text: of

text: the

text: eigenfunctions;

text: in

text: prac-

text: g

text: Ei

text: −1

text: −1

text: tice,

text: we

text: can

text: use

text: more

text: inducing

text: inputs

text: while

text: computing

text: only

text: the

text: top

text: M

text: eigenvectors.

text: As

text: shown

text: in

text: our

text: eigenfunction

text: x

text: x

text: lives

text: in

text: a

text: RKHS

text: as

text: a

text: linear

text: combination

text: of

text: the

text: kernel

text: func-

text: tions

text: evaluated

text: at

text: B

text: with

text: weights

text: u

text: j

text: Inserting

text: into

text: we

text: obtain

text: Figure

text: Illustration

text: of

text: the

text: effect

text: of

text: optimizing

text: basis

text: points

text: (i.e.,

text: inducing

text: inputs).

text: In

text: the

text: ﬁrst

text: row,

text: the

text: pink

text: dots

text: represent

text: data

text: points,

text: the

text: blue

text: and

text: red

text: solid

text: curves

text: correspond

text: to

text: the

text: M

text: M

text: predictive

text: mean

text: of

text: EigenGP

text: and

text: two

text: standard

text: deviations

text: f(x)

text: α

text: u

text: k(x,

text: b

text: j

text: ij

text: i

text: around

text: the

text: mean,

text: the

text: black

text: curve

text: corresponds

text: to

text: the

text: predic-

text: tive

text: mean

text: of

text: the

text: full

text: GP,

text: and

text: the

text: black

text: crosses

text: denote

text: the

text: basis

text: points.

text: In

text: the

text: second

text: row,

text: the

text: curves

text: in

text: various

text: colors

text: repre-

text: This

text: equation

text: reveals

text: a

text: two-layer

text: structure

text: of

text: EigenGP.

text: sent

text: the

text: ﬁve

text: eigenfunctions

text: of

text: EigenGP.

text: The

text: ﬁrst

text: layer

text: linearly

text: combines

text: multiple

text: kernel

text: functions

text: to

text: generate

text: each

text: eigenfunction

text: φj.

text: The

text: second

text: layer

text: takes

text: these

text: eigenfunctions

text: as

text: the

text: basis

text: functions

text: to

text: generate

text: the

text: function

text: ARD

text: kernel).

text: Thus,

text: this

text: model

text: can

text: accurately

text: model

text: the

text: un-

text: value

text: f.

text: Note

text: that

text: f

text: is

text: a

text: Bayesian

text: linear

text: combination

text: of

text: φj

text: certainty

text: of

text: a

text: test

text: point

text: even

text: when

text: it

text: is

text: far

text: from

text: the

text: training

text: where

text: the

text: weights

text: α

text: are

text: integrated

text: out

text: to

text: avoid

text: overﬁtting.

text: set.

text: We

text: derive

text: the

text: optimization

text: updates

text: of

text: all

text: the

text: hyperparam-

text: All

text: the

text: model

text: hyperparameters

text: are

text: learned

text: from

text: data.

text: eters

text: for

text: both

text: the

text: original

text: and

text: modiﬁed

text: EigenGP

text: models.

text: But

text: Speciﬁcally,

text: for

text: the

text: ﬁrst

text: layer,

text: to

text: learn

text: the

text: eigenfunctions

text: according

text: to

text: our

text: experiments,

text: the

text: modiﬁed

text: model

text: does

text: not

text: φ

text: we

text: estimate

text: the

text: inducing

text: inputs

text: B

text: and

text: the

text: kernel

text: hy-

text: i

text: improve

text: the

text: prediction

text: accuracy

text: over

text: the

text: original

text: EigenGP

text: (it

text: perparameters

text: (such

text: as

text: lengthscales

text: η

text: for

text: the

text: ARD

text: kernel)

text: by

text: even

text: reduces

text: the

text: accuracy

text: sometimes.).

text: Therefore,

text: we

text: will

text: maximizing

text: the

text: model

text: marginal

text: likelihood.

text: For

text: the

text: second

text: focus

text: on

text: the

text: original

text: EigenGP

text: model

text: in

text: our

text: presentation

text: for

text: layer,

text: we

text: marginalize

text: out

text: α

text: to

text: avoid

text: overﬁtting

text: and

text: maximize

text: its

text: simplicity.

text: Before

text: we

text: present

text: details

text: about

text: hyperparame-

text: the

text: model

text: marginal

text: likelihood

text: to

text: learn

text: the

text: hyperparameter

text: w

text: ter

text: optimization,

text: let

text: us

text: ﬁrst

text: look

text: at

text: an

text: illustrative

text: example

text: on

text: of

text: the

text: prior.

text: the

text: effect

text: of

text: hyperparameter

text: optimization,

text: in

text: particular,

text: the

text: With

text: the

text: estimated

text: hyperparameters,

text: the

text: prior

text: over

text: f

text: is

text: optimization

text: of

text: the

text: inducing

text: inputs

text: B.

text: nonstationary

text: because

text: its

text: covariance

text: function

text: in

text: varies

text: at

text: Illustrative

text: Example

text: different

text: regions

text: of

text: x.

text: This

text: comes

text: at

text: no

text: surprise

text: since

text: the

text: eigenfunctions

text: are

text: tied

text: with

text: p(x)

text: in

text: This

text: nonstationarity

text: For

text: this

text: example,

text: we

text: consider

text: a

text: toy

text: dataset

text: used

text: by

text: the

text: reﬂects

text: the

text: fact

text: that

text: our

text: model

text: is

text: adaptive

text: to

text: the

text: distribution

text: FITC

text: algorithm

text: [Snelson

text: and

text: Ghahramani,

text: It

text: contains

text: of

text: the

text: explanatory

text: variables

text: x.

text: one-dimensional

text: training

text: samples.

text: We

text: use

text: basis

text: points

text: Note

text: that

text: to

text: recover

text: the

text: full

text: uncertainty

text: captured

text: by

text: the

text: ker-

text: (M

text: and

text: choose

text: the

text: ARD

text: kernel.

text: We

text: then

text: compare

text: the

text: nel

text: function

text: k,

text: we

text: can

text: add

text: the

text: following

text: term

text: into

text: the

text: kernel

text: basis

text: functions

text: φj

text: and

text: the

text: corresponding

text: predictive

text: distri-

text: function

text: of

text: EigenGP:

text: butions

text: in

text: two

text: cases.

text: For

text: the

text: ﬁrst

text: case,

text: we

text: use

text: the

text: kernel

text: width

text: η

text: learned

text: from

text: the

text: full

text: GP

text: model

text: as

text: the

text: kernel

text: width

text: for

text: δ(x

text: k˜(x,

text: EigenGP,

text: and

text: apply

text: K-means

text: to

text: set

text: the

text: basis

text: points

text: B

text: as

text: clus-

text: ter

text: centers.

text: The

text: idea

text: of

text: using

text: K-means

text: to

text: set

text: the

text: basis

text: points

text: has

text: been

text: suggested

text: by

text: Zhang

text: et

text: al.

text: to

text: minimize

text: an

text: where

text: δ(a)

text: if

text: and

text: only

text: if

text: a

text: Compared

text: to

text: the

text: orig-

text: error

text: bound

text: for

text: the

text: Nystro¨m

text: approximation.

text: For

text: the

text: second

text: inal

text: EigenGP

text: model,

text: which

text: has

text: a

text: ﬁnite

text: degree

text: of

text: freedom,

text: case,

text: we

text: optimize

text: η,

text: B

text: and

text: w

text: by

text: maximizing

text: the

text: marginal

text: this

text: modiﬁed

text: model

text: has

text: the

text: inﬁnite

text: number

text: of

text: basis

text: functions

text: likelihood

text: of

text: EigenGP.

text: (assuming

text: k

text: has

text: an

text: inﬁnite

text: number

text: of

text: basis

text: functions

text: as

text: the

text: The

text: results

text: are

text: shown

text: in

text: Figure

text: The

text: ﬁrst

text: row

text: demon-

text: ##LTFigure##

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text: THEORETICAL ANALYSIS OF A STOCHASTIC APPROXIMATION APPROACH FOR COMPUTING QUASI-STATIONARY DISTRIBUTIONS JOSE BLANCHET, PETER GLYNN, SHUHENG ZHENG Abstract. This paper studies a method, which has been proposed in the Physics literature by [8, 7, 10], for estimating the quasi-stationary distribution. In contrast to existing methods in eigenvector estimation, the method eliminates the need for explicit transition matrix manipulation to extract the principal eigenvector. Our paper analyzes the algorithm by casting it as a stochastic approximation algorithm (Robbins-Monro) [23, 16]. In doing so, we prove its convergence and obtain its rate of convergence. Based on this insight, we also give an example where the rate of convergence is very slow. This problem can be alleviated by using an improved version of the algorithm that is given in this paper. Numerical experiments are described that demonstrate the eﬀectiveness of this improved method. 1. Introduction The motivation for this algorithm came from physicists’ need to estimate the quasi-stationary distribution of interacting particle systems (IPS) [8, 7, 10, 17]. A quasi-stationary distribution can be computed via the left principal eigenvector of the substochastic transition matrix over the non-absorbing states (transition rate matrix in continuous-time). However, the eigenvalue problem suﬀers from the curse of dimensionality, and is especially prohibitive in IPS where the state space is very large. A sampling based method has been proposed by these physicists based on a heuristic manipulation of the Kolmogorov forward equation. The validity of this method actually has been a priori proven by [1, 2, 21] who casted it as a generalized urn process. Results on rates of convergence have been obtained. This result involves a Central Limit Theorem (CLT) for a speciﬁc set of functionals corresponding to non-principal eigenvectors of the underlying substochastic matrix. Our main contribution are as follows (1) Our paper recognizes the algorithm as a stochastic approximation algorithm (Section ??). (2) This allows us to prove convergence and suﬃcient conditions for a stronger CLT (Theorem 3) that is not restricted only to speciﬁc functionals. (3) More importantly, we recognized common scenarios (Section ??) where the CLT fails and signiﬁcantly hamper the performance of the algorithm (i.e. very slow rate of convergence). (4) At the end, we came up with an improved algorithm (Section ??) which exhibits a valid CLT under all scenarios by using projection and iterate averaging [22]. Section 2 reviews some background material to the contact process, quasi-stationary distributions, mentions a less powerful method of proof via urn processes, and reviews the relevant related literature on eigenvector estimations and points out their shortcomings. Section 3 explains the the basis for the original heuristic and outlines the algorithm. Section 4 goes over the stochastic approximation formulation and sketches the proof of convergence (the full proof is given in the Appendix ??). Section 5 gives an improved version of the algorithm using projection along with its faster rate of convergence result. Section 6 brieﬂy studies the algorithm adapted for continuous-time Markov chains. Section 7 goes over several important numerical experiments. 1

text: THEORETICAL

text: ANALYSIS

text: OF

text: A

text: STOCHASTIC

text: APPROXIMATION

text: APPROACH

text: FOR

text: COMPUTING

text: QUASI-STATIONARY

text: DISTRIBUTIONS

text: JOSE

text: BLANCHET,

text: PETER

text: GLYNN,

text: SHUHENG

text: ZHENG

text: Abstract.

text: This

label: abstract

text: paper

text: studies

text: a

text: method,

text: which

text: has

text: been

text: proposed

text: in

text: the

text: Physics

text: literature

text: by

text: for

text: estimating

text: the

text: quasi-stationary

text: distribution.

text: In

text: contrast

text: to

text: existing

text: methods

text: in

text: eigenvector

text: estimation,

text: the

text: method

text: eliminates

text: the

text: need

text: for

text: explicit

text: transition

text: matrix

text: manipulation

text: to

text: extract

text: the

text: principal

text: eigenvector.

text: Our

text: paper

text: analyzes

text: the

text: algorithm

text: by

text: casting

text: it

text: as

text: a

text: stochastic

text: approximation

text: algorithm

text: (Robbins-Monro)

text: In

text: doing

text: so,

text: we

text: prove

text: its

text: convergence

text: and

text: obtain

text: its

text: rate

text: of

text: convergence.

text: Based

text: on

text: this

text: insight,

text: we

text: also

text: give

text: an

text: example

text: where

text: the

text: rate

text: of

text: convergence

text: is

text: very

text: slow.

text: This

text: problem

text: can

text: be

text: alleviated

text: by

text: using

text: an

text: improved

text: version

text: of

text: the

text: algorithm

text: that

text: is

text: given

text: in

text: this

text: paper.

text: Numerical

text: experiments

text: are

text: described

text: that

text: demonstrate

text: the

text: eﬀectiveness

text: of

text: this

text: improved

text: method.

text: Introduction

text: The

text: motivation

text: for

text: this

text: algorithm

text: came

text: from

text: physicists’

text: need

text: to

text: estimate

text: the

text: quasi-stationary

text: distribution

text: of

text: interacting

text: particle

text: systems

text: (IPS)

text: A

text: quasi-stationary

text: distribution

text: can

text: be

text: computed

text: via

text: the

text: left

text: principal

text: eigenvector

text: of

text: the

text: substochastic

text: transition

text: matrix

text: over

text: the

text: non-absorbing

text: states

text: (transition

text: rate

text: matrix

text: in

text: continuous-time).

text: However,

text: the

text: eigenvalue

text: problem

text: suﬀers

text: from

text: the

text: curse

text: of

text: dimensionality,

text: and

text: is

text: especially

text: prohibitive

text: in

text: IPS

text: where

text: the

text: state

text: space

text: is

text: very

text: large.

text: A

text: sampling

text: based

text: method

text: has

text: been

text: proposed

text: by

text: these

text: physicists

text: based

text: on

text: a

text: heuristic

text: manipulation

text: of

text: the

text: Kolmogorov

text: forward

text: equation.

text: The

text: validity

text: of

text: this

text: method

text: actually

text: has

text: been

text: a

text: priori

text: proven

text: by

text: who

text: casted

text: it

text: as

text: a

text: generalized

text: urn

text: process.

text: Results

text: on

text: rates

text: of

text: convergence

text: have

text: been

text: obtained.

text: This

text: result

text: involves

text: a

text: Central

text: Limit

text: Theorem

text: (CLT)

text: for

text: a

text: speciﬁc

text: set

text: of

text: functionals

text: corresponding

text: to

text: non-principal

text: eigenvectors

text: of

text: the

text: underlying

text: substochastic

text: matrix.

text: Our

text: main

text: contribution

text: are

text: as

text: follows

text: Our

text: paper

text: recognizes

text: the

text: algorithm

text: as

text: a

text: stochastic

text: approximation

text: algorithm

text: (Section

text: This

text: allows

text: us

text: to

text: prove

text: convergence

text: and

text: suﬃcient

text: conditions

text: for

text: a

text: stronger

text: CLT

text: (Theorem

text: that

text: is

text: not

text: restricted

text: only

text: to

text: speciﬁc

text: functionals.

text: More

text: importantly,

text: we

text: recognized

text: common

text: scenarios

text: (Section

text: where

text: the

text: CLT

text: fails

text: and

text: signiﬁcantly

text: hamper

text: the

text: performance

text: of

text: the

text: algorithm

text: (i.e.

text: very

text: slow

text: rate

text: of

text: convergence).

text: At

text: the

text: end,

text: we

text: came

text: up

text: with

text: an

text: improved

text: algorithm

text: (Section

text: which

text: exhibits

text: a

text: valid

text: CLT

text: under

text: all

text: scenarios

text: by

text: using

text: projection

text: and

text: iterate

text: averaging

text: Section

text: reviews

text: some

text: background

text: material

text: to

text: the

text: contact

text: process,

text: quasi-stationary

text: distributions,

text: mentions

text: a

text: less

text: powerful

text: method

text: of

text: proof

text: via

text: urn

text: processes,

text: and

text: reviews

text: the

text: relevant

text: related

text: literature

text: on

text: eigenvector

text: estimations

text: and

text: points

text: out

text: their

text: shortcomings.

text: Section

text: explains

text: the

text: the

text: basis

text: for

text: the

text: original

text: heuristic

text: and

text: outlines

text: the

text: algorithm.

text: Section

text: goes

text: over

text: the

text: stochastic

text: approximation

text: formulation

text: and

text: sketches

text: the

text: proof

text: of

text: convergence

text: (the

text: full

text: proof

text: is

text: given

text: in

text: the

text: Appendix

text: Section

text: gives

text: an

text: improved

text: version

text: of

text: the

text: algorithm

text: using

text: projection

text: along

text: with

text: its

text: faster

text: rate

text: of

text: convergence

text: result.

text: Section

text: brieﬂy

text: studies

text: the

text: algorithm

text: adapted

text: for

text: continuous-time

text: Markov

text: chains.

text: Section

text: goes

text: over

text: several

text: important

text: numerical

text: experiments.


text: STOCHASTIC APPROXIMATION FOR QUASI-STATIONARY DISTRIBUTIONS IN FINITE DIMENSIONS 11 Then there exists a unique quasi-stationary distribution µ satisfying the equations (cid:48) (cid:48) µ Q = λµ (cid:48) µ 1 = 1 ≥ (cid:0) (cid:1) µ 0 (cid:88) and the step sizes satisfy (cid:15) = Θ 1 for α < 0.5. We can conclude that the averaged sequence n nα n 1 ν = µ n n k k=1 converges to the point µ with probability 1. Furthermore, a strong CLT always hold √ − → n(ν µ) d N(0, V ) n for some covariance matrix V . 6. Algorithm for Continuous-Time Markov Chains So far, the exposition has assumed that the Markov chain of interest is a discrete-time process. It is straightforward to adapt our method for continuo(cid:20)us-time p(cid:21)rocesses (such as the contact process). If we denote the transition rate matrix of the CTMC in the following block form 0 0 T = N Q then we can write the algorithm as (cid:82) (cid:0) (cid:1) µ (x) = µ (x) + n+1 n (cid:80) 1 τn+1,m I(Xn+1(s) = x|Xn+1 ∼ µ ) ds − τn+1µ (x) (8) 0 (cid:0) (cid:1) 0 n n n + 1 1 n+1 τl n+1 l=0 1 − T = T + τn+1 T . n+1 n n + 2 n (cid:40) (cid:16) (cid:17) By a similar approach as the discrete-time case, we deduce the related dynamical system − (cid:48) − − (cid:48) − (cid:48) µ˙ (t) = 1 µ(t) Q 1 (µ(t) Q 11)µ(t) T(t) − (cid:48) − − T˙(t) = µ(t) Q 11 T(t). It is straightforward to adapt the Perron-Frobenius theorem to transition rate matrices such as Q by decom- − posing Q = A bI where A is an irreducible matrix. We know the existence of a principal eigenvector of positive entries µ¯ (with eigenvalue smaller than 0) such that (cid:48) (cid:48) µ¯ Q = λ¯µ¯ . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: STOCHASTIC

text: APPROXIMATION

text: FOR

text: QUASI-STATIONARY

text: DISTRIBUTIONS

text: IN

text: FINITE

text: DIMENSIONS

text: Then

text: there

text: exists

text: a

text: unique

text: quasi-stationary

text: distribution

text: µ

text: satisfying

text: the

text: equations

text: µ

text: Q

text: λµ

text: µ

text: µ

text: and

text: the

text: step

text: sizes

text: satisfy

text: Θ

text: for

text: α

text: We

text: can

text: conclude

text: that

text: the

text: averaged

text: sequence

text: n

text: nα

text: n

text: ν

text: µ

text: n

text: n

text: k

text: converges

text: to

text: the

text: point

text: µ

text: with

text: probability

text: Furthermore,

text: a

text: strong

text: CLT

text: always

text: hold

text: n(ν

text: µ)

text: d

text: V

text: n

text: for

text: some

text: covariance

text: matrix

text: V

label: section

text: Algorithm

text: for

text: Continuous-Time

text: Markov

text: Chains

text: So

text: far,

text: the

text: exposition

text: has

text: assumed

text: that

text: the

text: Markov

text: chain

text: of

text: interest

text: is

text: a

text: discrete-time

text: process.

text: It

text: is

text: straightforward

text: to

text: adapt

text: our

text: method

text: for

text: (such

text: as

text: the

text: contact

text: process).

text: If

text: we

text: denote

text: the

text: transition

text: rate

text: matrix

text: of

text: the

text: CTMC

text: in

text: the

text: following

text: block

text: form

text: T

text: N

text: Q

text: then

text: we

text: can

text: write

text: the

text: algorithm

text: as

text: µ

text: (x)

text: µ

text: (x)

text: n

text: µ

text: ds

text: (x)

text: n

text: n

text: n

text: τl

text: T

text: T

text: T

text: n

text: n

text: n

text: By

text: a

text: similar

text: approach

text: as

text: the

text: discrete-time

text: case,

text: we

text: deduce

text: the

text: related

text: dynamical

text: system

text: µ˙

text: (t)

text: µ(t)

text: Q

text: (µ(t)

text: Q

text: T(t)

text: T˙(t)

text: µ(t)

text: Q

text: T(t).

text: It

text: is

text: straightforward

text: to

text: adapt

text: the

text: Perron-Frobenius

text: theorem

text: to

text: transition

text: rate

text: matrices

text: such

text: as

text: Q

text: by

text: decom-

text: posing

text: Q

text: A

text: bI

text: where

text: A

text: is

text: an

text: irreducible

text: matrix.

text: We

text: know

text: the

text: existence

text: of

text: a

text: principal

text: eigenvector

text: of

text: positive

text: entries

text: µ¯

text: (with

text: eigenvalue

text: smaller

text: than

text: such

text: that

text: µ¯

text: Q

text: λ¯µ¯

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##