text: 2.3 Rectiﬁed Gaussian model The DSP model assumes that the ﬁring rate deﬁnes the probability with which spikes are generated in a given time window. Inspired by the relatively little stochasticity found in sensory neurons in vitro (Mainen and Sejnowski, 1995), an alternative model can be formulated, which assumes a deterministic process for spike generation (Carandini, 2004). Self-consistency requires that the number of spikes with constant stimulus on average is proportional to the ﬁring rate. A model that assumes no further source of variability but fulﬁls the self consistency criterion can be formulated by integrating the ﬁring rate over time and generating spikes whenever the integral crosses integer values. If we consider the rate to be constant within a time bin b, scaled appropriately with (cid:88) the base rate parameter k, the integral becomes a ﬁnite sum, for neuron n: (cid:98) τ (cid:99) sn = rn (4) t b b=1 The spiking model deﬁned this way, the Rectiﬁed Gaussian model (RG), is formally equivalent to an integrate-and-ﬁre neuron model without refractory period with the addition of the ﬁring rate nonlinearity. The principal source of variability in the RG model is coming from the variability present in the membrane potentials. An additional, though minor, source of variability for the timing of spikes originates from an uncertainty of the state of the integrator at the beginning of a trial. A cartoon depicting the procedure of generating ﬁring rates from Gaussian membrane potentials, and then spike counts using the two models, is summa- rized in Fig. 1. 2.4 Simulation of population measurements In order to compare the predictions of the RG and DSP models with ex- perimental recordings of activity of a population of neurons, population models were constructed and spike response statistics of the two models were contrasted with experimental data. The primary goal of these comparisons was to test pre- dictions of the two models on the population level, where summary statistics are less susceptible to sampling noise, and their measurements do not require the precise control of receptive ﬁeld contents. We simulated full-ﬁeld gratings as stimuli in the simulated experimental paradigm. Determining the membrane potential statistics for the population of N neu- rons requires the speciﬁcation of their means, variances and correlations, as speciﬁed in Eq. 1. The mean membrane potential response µn for cell n is de- termined by stimulus orientation through the tuning curve that is characteristic to the particular neuron. The stimulus orientation at which the peak of the tuning curve is located corresponds to the preferred orientation of the neuron and preferred orientations are sampled independently for each neuron from a uniform distribution. The height of the tuning curve is varying from neuron to neuron and is sampled from a Gaussian distribution with a standard devi- 6

text: Rectiﬁed

text: Gaussian

text: model

text: The

text: DSP

text: model

text: assumes

text: that

text: the

text: ﬁring

text: rate

text: deﬁnes

text: the

text: probability

text: with

text: which

text: spikes

text: are

text: generated

text: in

text: a

text: given

text: time

text: window.

text: Inspired

text: by

text: the

text: relatively

text: little

text: stochasticity

text: found

text: in

text: sensory

text: neurons

text: in

text: vitro

text: (Mainen

text: and

text: Sejnowski,

text: an

text: alternative

text: model

text: can

text: be

text: formulated,

text: which

text: assumes

text: a

text: deterministic

text: process

text: for

text: spike

text: generation

text: (Carandini,

text: Self-consistency

text: requires

text: that

text: the

text: number

text: of

text: spikes

text: with

text: constant

text: stimulus

text: on

text: average

text: is

text: proportional

text: to

text: the

text: ﬁring

text: rate.

text: A

text: model

text: that

text: assumes

text: no

text: further

text: source

text: of

text: variability

text: but

text: fulﬁls

text: the

text: self

text: consistency

text: criterion

text: can

text: be

text: formulated

text: by

text: integrating

text: the

text: ﬁring

text: rate

text: over

text: time

text: and

text: generating

text: spikes

text: whenever

text: the

text: integral

text: crosses

text: integer

text: values.

text: If

text: we

text: consider

text: the

text: rate

text: to

text: be

text: constant

text: within

text: a

text: time

text: bin

text: b,

text: scaled

text: appropriately

text: with

text: the

text: base

text: rate

text: parameter

text: k,

text: the

text: integral

text: becomes

text: a

text: ﬁnite

text: sum,

text: for

text: neuron

text: n:

text: τ

text: sn

text: rn

text: t

text: b

text: The

text: spiking

text: model

text: deﬁned

text: this

text: way,

text: the

text: Rectiﬁed

text: Gaussian

text: model

text: (RG),

text: is

text: formally

text: equivalent

text: to

text: an

text: integrate-and-ﬁre

text: neuron

text: model

text: without

text: refractory

text: period

text: with

text: the

text: addition

text: of

text: the

text: ﬁring

text: rate

text: nonlinearity.

text: The

text: principal

text: source

text: of

text: variability

text: in

text: the

text: RG

text: model

text: is

text: coming

text: from

text: the

text: variability

text: present

text: in

text: the

text: membrane

text: potentials.

text: An

text: additional,

text: though

text: minor,

text: source

text: of

text: variability

text: for

text: the

text: timing

text: of

text: spikes

text: originates

text: from

text: an

text: uncertainty

text: of

text: the

text: state

text: of

text: the

text: integrator

text: at

text: the

text: beginning

text: of

text: a

text: trial.

text: A

text: cartoon

text: depicting

text: the

text: procedure

text: of

text: generating

text: ﬁring

text: rates

text: from

text: Gaussian

text: membrane

text: potentials,

text: and

text: then

text: spike

text: counts

text: using

text: the

text: two

text: models,

text: is

text: summa-

text: rized

text: in

text: Fig.

text: Simulation

text: of

text: population

text: measurements

text: In

text: order

text: to

text: compare

text: the

text: predictions

text: of

text: the

text: RG

text: and

text: DSP

text: models

text: with

text: ex-

text: perimental

text: recordings

text: of

text: activity

text: of

text: a

text: population

text: of

text: neurons,

text: population

text: models

text: were

text: constructed

text: and

text: spike

text: response

text: statistics

text: of

text: the

text: two

text: models

text: were

text: contrasted

text: with

text: experimental

text: data.

text: The

text: primary

text: goal

text: of

text: these

text: comparisons

text: was

text: to

text: test

text: pre-

text: dictions

text: of

text: the

text: two

text: models

text: on

text: the

text: population

text: level,

text: where

text: summary

text: statistics

text: are

text: less

text: susceptible

text: to

text: sampling

text: noise,

text: and

text: their

text: measurements

text: do

text: not

text: require

text: the

text: precise

text: control

text: of

text: receptive

text: ﬁeld

text: contents.

text: We

text: simulated

text: full-ﬁeld

text: gratings

text: as

text: stimuli

text: in

text: the

text: simulated

text: experimental

text: paradigm.

text: Determining

text: the

text: membrane

text: potential

text: statistics

text: for

text: the

text: population

text: of

text: N

text: neu-

text: rons

text: requires

text: the

text: speciﬁcation

text: of

text: their

text: means,

text: variances

text: and

text: correlations,

text: as

text: speciﬁed

text: in

text: Eq.

text: The

text: mean

text: membrane

text: potential

text: response

text: µn

text: for

text: cell

text: n

text: is

text: de-

text: termined

text: by

text: stimulus

text: orientation

text: through

text: the

text: tuning

text: curve

text: that

text: is

text: characteristic

text: to

text: the

text: particular

text: neuron.

text: The

text: stimulus

text: orientation

text: at

text: which

text: the

text: peak

text: of

text: the

text: tuning

text: curve

text: is

text: located

text: corresponds

text: to

text: the

text: preferred

text: orientation

text: of

text: the

text: neuron

text: and

text: preferred

text: orientations

text: are

text: sampled

text: independently

text: for

text: each

text: neuron

text: from

text: a

text: uniform

text: distribution.

text: The

text: height

text: of

text: the

text: tuning

text: curve

text: is

text: varying

text: from

text: neuron

text: to

text: neuron

text: and

text: is

text: sampled

text: from

text: a

text: Gaussian

text: distribution

text: with

text: a

text: standard

text: devi-


text: 4 for two qubits in the computational basis. Bell GHZ Combining this deﬁnition of extended parity with the qubit 1 2 0 1 2 3 3 U composite rotation operator, N we obtain the kernel T1(µs) 85.8 75.1 58.9 87.1 74.7 74.8 65.5 ∆ ({θ , ϕ }) = U Πˆ U† (8) T(cid:15) 2((×µs1)0−2) 1009..615 580..82 740..829 1420..22 590..223 530..223 480..489 SU(2[N]) i i N SU(2[N]) N g × − (cid:15) ( 10 2) 4.6 4.3 4.6 4.2 3.6 3.6 5.7 r × − that satisﬁes the restricted Stratonovich-Weyl correspon- (cid:15)i2( 10 2) 3.19 5.21 3.31 3.18 6.55 g (cid:0) (cid:1) dence given in the introduction. We note that the Φ ’s i make no contribution as Π commutes with σ . TABLE I. Calibration data for the experimental results con- SU(2[N]) zi tained within this paper. Data for the Bell state and GHZ This kernel deﬁnes our SU 2[(cid:104)N] , extended pari(cid:105)ty-based, Wigner functions were taken on 16th and 17th June 2016 when Wigner function according to the fridge temperature was 18.25 mK and 17.916 mK respec- U U† tively. T1 and T2 are the usual relaxation times, (cid:15)g is the gate WSU(2[N])({θi, ϕi}) = Tr ρ NΠˆSU(2[N]) N . (9) error, (cid:15)r is the readout error and (cid:15)ig2 is the C-NOT gate er- ror between the qubit listed and qubit 2 (which is the target Let us now consider the speciﬁc case of the Wigner qubit for the C-NOT operation). function W for two qubits. Each qubit brings with SU(2[N]) it two degrees of freedom, expressed in terms of Euler angles Ω = (θ , ϕ , θ , ϕ ), thus the associated Wigner of points as needed to reconstruct the density matrix. 1 1 2 2 In Fig. 3 we have actually measured more, and diﬀer- function takes the form of a four-dimensional pseudo- probability distribution W (θ , ϕ , θ , ϕ ). Four- ent, points than would be needed to fully reconstruct SU(2[2]) 1 1 2 2 the state. This was done to demonstrate the ability to dimensional functions are not easy to visualise, but we generate the Wigner function using a raster scan ap- can take slices of the function in order to gain an appre- proach as this makes clear the straightforward nature ciation of it as a whole. In Fig. 2 (a-d) we show some ex- of our measurement method. Due to ﬁnite computa- ample Wigner function slices for two Bell states. Specif- (cid:94) tional resources, and the need to do rotations as outlined ically Fig. 2 (a,b) shows the equal angle (“= ”) slice (cid:94) above, we are limited in our resolution. Nevertheless, W= (θ, ϕ) = W (θ, ϕ, θ, ϕ) while Fig. 2 (c,d) SU(2[2]) SU(2[2]) we ﬁnd good agreement between theory, simulation, and shows the slice Wϕi=0 (θ , θ ) = W (θ , 0, θ , 0). experimental data, demonstrating that our tomographic SU(2[2]) 1 2 SU(2[2]) 1 2 process is clearly able to distinguish between the two Note that Fig. 2 (e,f) will be discussed in section V. Bell states. In order to demonstrate that this function is indeed Bell states are interesting both as an example of easy to construct we have taken advantage of IBM’s maximally entangled states and for their usefulness in Quantum Experience project. The project makes avail- quantum information processing. Fortunately, for sys- able through the Internet a ﬁve-qubit processor, initially tems comprising more spins, we can extend this class of based on a simple “star” topology3: a central qubit states to those that have a direct analogy with optical is coupled to four other qubits. The machine has al- Schr¨odinger cat states as considered in [5] and others. ready been used to produce interesting results [47, 48]. Such states are termed “spin-cat states” of which the Here we use it to measure and reconstruct the Wigner GHZ state [49] is an excellent example. In previous the- functions for the two Bell states |Φ (cid:105) and |Ψ−(cid:105) as pre- + oretical work, spin Wigner-like functions have been pro- sented in Fig. 2. In this work, we are limited by the posed as a mechanism for visualizing such cat states [13, operations that IBM has made available to the user, 50, 51]. In analogy with measuring Wigner functions operations that naturally focus on quantum computing of non-classical cavity ﬁeld states [5], using our method applications. Nevertheless, following Eq. (2), we are able we now construct the W Wigner function for a to produce ρ˜(θ , ϕ , θ , ϕ ) using rotations generated by SU(2[5]) 1 1 2 2 spin-cat of the form |GHZ (cid:105) = (|0(cid:105) |0(cid:105) |0(cid:105) |0(cid:105) |0(cid:105) + combinations of gate operations and readout state pop- √ 5 1 2 3 4 5 ulations of ρ˜nn(θ1, ϕ1, θ2, ϕ2) via the standard output of |1(cid:105)1 |1(cid:105)2 |1(cid:105)3 |1(cid:105)4 |1(cid:105)5)/ 2. In Fig. 4 we show the θ1 = the IBM processor. We then use Eq. (4) and Eq. (7) to θ2 = θ3 = θ4 = θ5 and ϕ1 = ϕ2 = ϕ3 = ϕ4 = ϕ5 slice reconstruct the Wigner function, Eq. (9). of the WSU(2[5]) Wigner function for |GHZ5(cid:105) which is the In Fig. 3 we plot the Wigner function Wϕi=0 (θ , θ ) higher dimensional analogue of Fig. 2 (a,b). We show SU(2[2]) 1 2 both theoretical predictions and, due to limited compu- slices comparing the ideal theoretical values of Fig. 2 tational resource, as insets, simulation and experimental (c,d), values generated by IBM’s built in simulator (that data obtained from the IBM machine. Once more the cal- models environmental eﬀects), and real experimental ibration data pertaining to the experiments is provided data. The calibration data pertaining to the experi- in Table I. We note that the θ = θ = θ = θ = θ ments is provided in Table I. In principle, to fully recon- 1 2 3 4 5 and ϕ = ϕ = ϕ = ϕ = ϕ slice does not contain all struct the state requires us to measure the same number 1 2 3 4 5 the information needed to reconstruct the state; for full reconstruction we would need to measure and visualise all {θ , θ } i (cid:54)= j sets of angles for various values of ϕ . i j i 3 Before the early 2017 update by IBM For the top and bottom point the theoretical value is ##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##

label: paragraph

text: for

text: two

text: qubits

text: in

text: the

text: computational

text: basis.

text: Bell

label: table

text: GHZ

text: Combining

text: this

text: deﬁnition

text: of

text: extended

text: parity

text: with

text: the

text: qubit

text: U

text: composite

text: rotation

text: operator,

text: N

text: we

text: obtain

text: the

text: kernel

text: ({θ

text: ϕ

text: U

text: Πˆ

text: U†

text: i

text: i

text: N

text: N

text: g

text: r

text: that

text: satisﬁes

text: the

text: restricted

text: Stratonovich-Weyl

text: correspon-

text: g

text: dence

text: given

text: in

text: the

text: introduction.

text: We

text: note

text: that

text: the

text: Φ

text: ’s

text: i

text: make

text: no

text: contribution

text: as

text: Π

text: commutes

text: with

text: σ

text: TABLE

text: I.

text: Calibration

text: data

text: for

text: the

text: experimental

text: results

text: con-

text: zi

text: tained

text: within

text: this

text: paper.

text: Data

text: for

text: the

text: Bell

text: state

text: and

text: GHZ

text: This

text: kernel

text: deﬁnes

text: our

text: SU

text: extended

text: Wigner

text: functions

text: were

text: taken

text: on

text: and

text: June

text: when

text: Wigner

text: function

text: according

text: to

text: the

text: fridge

text: temperature

text: was

text: mK

text: and

text: mK

text: respec-

text: U

text: U†

text: tively.

text: and

text: are

text: the

text: usual

text: relaxation

text: times,

text: is

text: the

text: gate

text: ϕi})

text: Tr

text: ρ

text: N

text: error,

text: is

text: the

text: readout

text: error

text: and

text: is

text: the

text: C-NOT

text: gate

text: er-

text: ror

text: between

text: the

text: qubit

text: listed

text: and

text: qubit

text: (which

text: is

text: the

text: target

text: Let

text: us

text: now

text: consider

text: the

text: speciﬁc

text: case

text: of

text: the

text: Wigner

text: qubit

text: for

text: the

text: C-NOT

text: operation).

text: function

text: W

text: for

text: two

text: qubits.

text: Each

text: qubit

text: brings

text: with

text: it

text: two

text: degrees

text: of

text: freedom,

text: expressed

text: in

text: terms

text: of

text: Euler

text: angles

text: Ω

text: (θ

text: ϕ

text: θ

text: ϕ

text: thus

text: the

text: associated

text: Wigner

text: of

text: points

text: as

text: needed

text: to

text: reconstruct

text: the

text: density

text: matrix.

text: In

text: Fig.

text: we

text: have

text: actually

text: measured

text: more,

text: and

text: diﬀer-

text: function

text: takes

text: the

text: form

text: of

text: a

text: four-dimensional

text: pseudo-

text: probability

text: distribution

text: W

text: (θ

text: ϕ

text: θ

text: ϕ

text: Four-

text: ent,

text: points

text: than

text: would

text: be

text: needed

text: to

text: fully

text: reconstruct

text: the

text: state.

text: This

text: was

text: done

text: to

text: demonstrate

text: the

text: ability

text: to

text: dimensional

text: functions

text: are

text: not

text: easy

text: to

text: visualise,

text: but

text: we

text: generate

text: the

text: Wigner

text: function

text: using

text: a

text: raster

text: scan

text: ap-

text: can

text: take

text: slices

text: of

text: the

text: function

text: in

text: order

text: to

text: gain

text: an

text: appre-

text: proach

text: as

text: this

text: makes

text: clear

text: the

text: straightforward

text: nature

text: ciation

text: of

text: it

text: as

text: a

text: whole.

text: In

text: Fig.

text: (a-d)

text: we

text: show

text: some

text: ex-

text: of

text: our

text: measurement

text: method.

text: Due

text: to

text: ﬁnite

text: computa-

text: ample

text: Wigner

text: function

text: slices

text: for

text: two

text: Bell

text: states.

text: Specif-

text: tional

text: resources,

text: and

text: the

text: need

text: to

text: do

text: rotations

text: as

text: outlined

text: ically

text: Fig.

text: (a,b)

text: shows

text: the

text: equal

text: angle

text: slice

text: above,

text: we

text: are

text: limited

text: in

text: our

text: resolution.

text: Nevertheless,

text: W=

text: (θ,

text: ϕ)

text: W

text: (θ,

text: ϕ,

text: θ,

text: ϕ)

text: while

text: Fig.

text: (c,d)

text: we

text: ﬁnd

text: good

text: agreement

text: between

text: theory,

text: simulation,

text: and

text: shows

text: the

text: slice

text: (θ

text: θ

text: W

text: (θ

text: θ

text: experimental

text: data,

text: demonstrating

text: that

text: our

text: tomographic

text: process

text: is

text: clearly

text: able

text: to

text: distinguish

text: between

text: the

text: two

text: Note

text: that

text: Fig.

text: (e,f)

text: will

text: be

text: discussed

text: in

text: section

text: V.

text: Bell

text: states.

text: In

text: order

text: to

text: demonstrate

text: that

text: this

text: function

text: is

text: indeed

text: Bell

text: states

text: are

text: interesting

text: both

text: as

text: an

text: example

text: of

text: easy

text: to

text: construct

text: we

text: have

text: taken

text: advantage

text: of

text: IBM’s

text: maximally

text: entangled

text: states

text: and

text: for

text: their

text: usefulness

text: in

text: Quantum

text: Experience

text: project.

text: The

text: project

text: makes

text: avail-

text: quantum

text: information

text: processing.

text: Fortunately,

text: for

text: sys-

text: able

text: through

text: the

text: Internet

text: a

text: ﬁve-qubit

text: processor,

text: initially

text: tems

text: comprising

text: more

text: spins,

text: we

text: can

text: extend

text: this

text: class

text: of

text: based

text: on

text: a

text: simple

text: “star”

text: a

text: central

text: qubit

text: states

text: to

text: those

text: that

text: have

text: a

text: direct

text: analogy

text: with

text: optical

text: is

text: coupled

text: to

text: four

text: other

text: qubits.

text: The

text: machine

text: has

text: al-

text: Schr¨odinger

text: cat

text: states

text: as

text: considered

text: in

text: and

text: others.

text: ready

text: been

text: used

text: to

text: produce

text: interesting

text: results

text: Such

text: states

text: are

text: termed

text: “spin-cat

text: states”

text: of

text: which

text: the

text: Here

text: we

text: use

text: it

text: to

text: measure

text: and

text: reconstruct

text: the

text: Wigner

text: GHZ

text: state

text: is

text: an

text: excellent

text: example.

text: In

text: previous

text: the-

text: functions

text: for

text: the

text: two

text: Bell

text: states

text: |Φ

text: and

text: as

text: pre-

text: oretical

text: work,

text: spin

text: Wigner-like

text: functions

text: have

text: been

text: pro-

text: sented

text: in

text: Fig.

text: In

text: this

text: work,

text: we

text: are

text: limited

text: by

text: the

text: posed

text: as

text: a

text: mechanism

text: for

text: visualizing

text: such

text: cat

text: states

text: operations

text: that

text: IBM

text: has

text: made

text: available

text: to

text: the

text: user,

text: In

text: analogy

text: with

text: measuring

text: Wigner

text: functions

text: operations

text: that

text: naturally

text: focus

text: on

text: quantum

text: computing

text: of

text: non-classical

text: cavity

text: ﬁeld

text: states

text: using

text: our

text: method

text: applications.

text: Nevertheless,

text: following

text: Eq.

text: we

text: are

text: able

text: we

text: now

text: construct

text: the

text: W

text: Wigner

text: function

text: for

text: a

text: to

text: produce

text: ρ˜(θ

text: ϕ

text: θ

text: ϕ

text: using

text: rotations

text: generated

text: by

text: spin-cat

text: of

text: the

text: form

text: |GHZ

text: combinations

text: of

text: gate

text: operations

text: and

text: readout

text: state

text: pop-

text: ulations

text: of

text: via

text: the

text: standard

text: output

text: of

text: In

text: Fig.

text: we

text: show

text: the

text: θ1

text: the

text: IBM

text: processor.

text: We

text: then

text: use

text: Eq.

text: and

text: Eq.

text: to

text: θ2

text: θ3

text: θ4

text: θ5

text: and

text: ϕ1

text: ϕ2

text: ϕ3

text: ϕ4

text: ϕ5

text: slice

text: reconstruct

text: the

text: Wigner

text: function,

text: Eq.

text: of

text: the

text: Wigner

text: function

text: for

text: which

text: is

text: the

text: In

text: Fig.

text: we

text: plot

text: the

text: Wigner

text: function

text: (θ

text: θ

text: higher

text: dimensional

text: analogue

text: of

text: Fig.

text: (a,b).

text: We

text: show

text: both

text: theoretical

text: predictions

text: and,

text: due

text: to

text: limited

text: compu-

text: slices

text: comparing

text: the

text: ideal

text: theoretical

text: values

text: of

text: Fig.

text: tational

text: resource,

text: as

text: insets,

text: simulation

text: and

text: experimental

text: (c,d),

text: values

text: generated

text: by

text: IBM’s

text: built

text: in

text: simulator

text: (that

text: data

text: obtained

text: from

text: the

text: IBM

text: machine.

text: Once

text: more

text: the

text: cal-

text: models

text: environmental

text: eﬀects),

text: and

text: real

text: experimental

text: ibration

text: data

text: pertaining

text: to

text: the

text: experiments

text: is

text: provided

text: data.

text: The

text: calibration

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text: to

text: the

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text: in

text: Table

text: I.

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text: note

text: that

text: the

text: θ

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text: I.

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text: principle,

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text: all

text: {θ

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text: 7 This leads to a extended parity operator that is nothing more than the tensor product of the parities of individual qubits: (cid:79) (cid:79) Simulation 2 (cid:78) Experiment Π = N Π(i) = N 1(1l + √3σ ), (10) 1 N SU(2) SU(2[1]) 2 zi i=1 i=1 0 which for one qubit is equal to Eq. (6) but for two qubits takes the explicit form    √  (cid:78) 2 + 3 0 0 0  1 0 −1 0 0 Π = (11) 2 SU(2) 2 0 0 −1 0√ 0 0 0 2 − 3 Simulation 1 √ √ Experiment = [1l ⊗ 1l + 3 1l ⊗ σ + 3 σ ⊗ 1l + 3 σ ⊗ σ ] z z z z 4 Simulation Simulation in the computational basis. When compared with Eq. (7) we see that this version of extended parity no longer Experiment Experiment treats one-qubit and two-qubit contributions on an equal footing. The deﬁnition of the Wigner function continues in the same way as before(cid:104) and, in terms (cid:105)of the rotated U† U Theory Simulated Experimental density matrix ρ˜ = ρ takes the form (cid:78) N(cid:88)N (cid:78)(cid:16) (cid:17) |Φ−(cid:105) W (Ω) = Tr ρ˜(Ω)Π N SU(2) N S(cid:78)U(2) θ θ θ 2 2 2 = ρ˜nn(Ω) Π N SU(2) . (12) 1 nn n θ θ θ 1 1 1 Re(cid:78)turning to Fig. 2 (e(cid:78),f) we show example slices of | (cid:105) 0.5 Ψ Wθi=0 (θ , θ ) = W (θ , ϕ = 0, θ , ϕ = 0) + N SU(2) 1 2 N SU(2) 1 1 2 2 θ θ θ that demonstrates this alternative Wigner function is 2 2 2 0 qualitatively very similar to the equivalent slices of W (Ω) shown in Fig. 2 (c,d). − SU(2[N]) θ1 θ1 θ1 0.5 In Fig. 5 (top) we show results for comparison with Fig. 4 and (bottom) (and, by analogy, with non-classical cav- (cid:78) FIG. 5. Here we reproduce Figs. 3 and 4 using the same ity ﬁeld states [5]) with Fig. 3 which demonstrates that data but now employing the Wigner function deﬁned using W is a Wigner function with qualitatively very N SU(2) the alternative extended parity operators as given in Eq. (12). similar features to W that will be compared in SU(2[N]) In the top ﬁgure, for comparison with Fig. 4, we show the (cid:78) the next section. For the top and bottom point the theo- ﬁve-qubit GHZ spin Schro¨dinger cat state Wigner function retical value is 2.375. The simulated values are 1.13 and W 5 SU(2) for the θ1 = θ2 = θ3 = θ4 = θ5 and ϕ1 = ϕ2 = 1.11, and the experimental values are 0.8876 and 0.9006, ϕ3 = ϕ4 = ϕ5 slice. Again we show the ideal function, and as insets, show both simulated and experimental results from respectively. (cid:78) IBM’s Quantum Experience project. On the bottom ﬁgure, VI. EFFICIENT STATE ESTIMATION, for comparison with Fig. 3, we provide plots of W for CHARACTERISATION, AND ENTANGLEMENT | (cid:105) | (cid:105) 2 SU(2) the two Bell states Φ and Ψ− . We plot θ versus θ for VALIDATION + 1 2 the ϕ = ϕ = 0 slice of the Wigner function for two qubits. 1 2 Once more, we have included for comparison ideal theoreti- As they are informationally complete, our Wigner cal values, numerical results using IBM’s built in simulator, functions for spin can be considered mathematically and real experimental data from IBM’s quantum processor. equivalent to the density matrix/state space formula- Again we see good agreement between theory, simulation and tion. They also exhibit unique, and intuitively natural, experiment and note that using a diﬀerent extended parity characteristic features. If, for example, we look at Fig. 5 operator provides an alternative path to direct measurement for the GHZ state (which is a superposition of spin co- of phase space. herent states) it is clear that there are regions of strong oscillations in the equal angle slice; these are reminiscent of the interference terms between two harmonic oscil- used to verify non-classical properties of the state such lator coherent states shown in Fig. 1. It is natural to as quantum coherence or entanglement. In other words, ask if measurement of such characteristic features can be can we extract information in a similar way as for Wigner ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: leads

text: to

text: a

text: extended

text: parity

text: operator

text: that

text: is

text: nothing

text: more

text: than

text: the

text: tensor

text: product

text: of

text: the

text: parities

text: of

text: individual

text: qubits:

text: Simulation

text: Experiment

text: Π

text: N

text: Π(i)

text: N

text: √3σ

text: N

text: zi

text: which

text: for

text: one

text: qubit

text: is

text: equal

text: to

text: Eq.

text: but

text: for

text: two

text: qubits

text: takes

text: the

text: explicit

text: form

text: 2

text: −1

text: Π

text: −1

text: 0√

text: Simulation

text: Experiment

text: σ

text: σ

text: σ

text: σ

text: z

text: z

text: z

text: z

text: Simulation

text: Simulation

text: in

text: the

text: computational

text: basis.

text: When

text: compared

text: with

text: Eq.

text: we

text: see

text: that

text: this

text: version

text: of

text: extended

text: parity

text: no

text: longer

text: Experiment

text: Experiment

text: treats

text: one-qubit

text: and

text: two-qubit

text: contributions

text: on

text: an

text: equal

text: footing.

text: The

text: deﬁnition

text: of

text: the

text: Wigner

text: function

text: continues

text: in

text: the

text: same

text: way

text: as

text: and,

text: in

text: terms

text: the

text: rotated

text: U†

text: U

text: Theory

text: Simulated

text: Experimental

text: density

text: matrix

text: ρ˜

text: ρ

text: takes

text: the

text: form

text: W

text: (Ω)

text: Tr

text: ρ˜(Ω)Π

text: N

text: N

text: θ

text: θ

text: θ

text: ρ˜nn(Ω)

text: Π

text: N

text: nn

text: n

text: θ

text: θ

text: θ

text: to

text: Fig.

text: we

text: show

text: example

text: slices

text: of

text: Ψ

text: (θ

text: θ

text: W

text: (θ

text: ϕ

text: θ

text: ϕ

text: N

text: N

text: θ

text: θ

text: θ

text: that

text: demonstrates

text: this

text: alternative

text: Wigner

text: function

text: is

text: qualitatively

text: very

text: similar

text: to

text: the

text: equivalent

text: slices

text: of

text: W

text: (Ω)

text: shown

text: in

text: Fig.

text: (c,d).

text: θ1

text: θ1

text: θ1

text: In

text: Fig.

text: (top)

text: we

text: show

text: results

text: for

text: comparison

text: with

text: Fig.

text: and

text: (bottom)

text: (and,

text: by

text: analogy,

text: with

text: non-classical

text: cav-

text: FIG.

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text: we

text: reproduce

text: Figs.

text: and

text: using

text: the

text: same

text: ity

text: ﬁeld

text: states

text: with

text: Fig.

text: which

text: demonstrates

text: that

text: data

text: but

text: now

text: employing

text: the

text: Wigner

text: function

text: deﬁned

text: using

text: W

text: is

text: a

text: Wigner

text: function

text: with

text: qualitatively

text: very

text: N

text: the

text: alternative

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text: parity

text: operators

text: as

text: given

text: in

text: Eq.

text: similar

text: features

text: to

text: W

text: that

text: will

text: be

text: compared

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text: In

text: the

text: top

text: ﬁgure,

text: for

text: comparison

text: with

text: Fig.

text: we

text: show

text: the

text: the

text: next

text: section.

text: For

text: the

text: top

text: and

text: bottom

text: point

text: the

text: theo-

text: ﬁve-qubit

text: GHZ

text: spin

text: Schro¨dinger

text: cat

text: state

text: Wigner

text: function

text: retical

text: value

text: is

text: The

text: simulated

text: values

text: are

text: and

text: W

text: for

text: the

text: θ1

text: θ2

text: θ3

text: θ4

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text: ϕ1

text: ϕ2

text: and

text: the

text: experimental

text: values

text: are

text: and

text: ϕ3

text: ϕ4

text: ϕ5

text: slice.

text: Again

text: we

text: show

text: the

text: ideal

text: function,

text: and

text: as

text: insets,

text: show

text: both

text: simulated

text: and

text: experimental

text: results

text: from

text: respectively.

text: IBM’s

text: Quantum

text: Experience

text: project.

text: On

text: the

text: bottom

text: ﬁgure,

text: VI.

text: EFFICIENT

text: STATE

text: ESTIMATION,

text: for

text: comparison

text: with

text: Fig.

text: we

text: provide

text: plots

text: of

text: W

text: for

text: CHARACTERISATION,

text: AND

text: ENTANGLEMENT

text: the

text: two

text: Bell

text: states

text: Φ

text: and

text: Ψ−

text: We

text: plot

text: θ

text: versus

text: θ

text: for

text: VALIDATION

text: the

text: ϕ

text: ϕ

text: slice

text: of

text: the

text: Wigner

text: function

text: for

text: two

text: qubits.

text: Once

text: more,

text: we

text: have

text: included

text: for

text: comparison

text: ideal

text: theoreti-

text: As

text: they

text: are

text: informationally

text: complete,

text: our

text: Wigner

text: cal

text: values,

text: numerical

text: results

text: using

text: IBM’s

text: built

text: in

text: simulator,

text: functions

text: for

text: spin

text: can

text: be

text: considered

text: mathematically

text: and

text: real

text: experimental

text: data

text: from

text: IBM’s

text: quantum

text: processor.

text: equivalent

text: to

text: the

text: density

text: matrix/state

text: space

text: formula-

text: Again

text: we

text: see

text: good

text: agreement

text: between

text: theory,

text: simulation

text: and

text: tion.

text: They

text: also

text: exhibit

text: unique,

text: and

text: intuitively

text: natural,

text: experiment

text: and

text: note

text: that

text: using

text: a

text: diﬀerent

text: extended

text: parity

text: characteristic

text: features.

text: If,

text: for

text: example,

text: we

text: look

text: at

text: Fig.

text: operator

text: provides

text: an

text: alternative

text: path

text: to

text: direct

text: measurement

text: for

text: the

text: GHZ

text: state

text: (which

text: is

text: a

text: superposition

text: of

text: spin

text: co-

text: of

text: phase

text: space.

text: herent

text: states)

text: it

text: is

text: clear

text: that

text: there

text: are

text: regions

text: of

text: strong

text: oscillations

text: in

text: the

text: equal

text: angle

text: slice;

text: these

text: are

text: reminiscent

text: of

text: the

text: interference

text: terms

text: between

text: two

text: harmonic

text: oscil-

text: used

text: to

text: verify

text: non-classical

text: properties

text: of

text: the

text: state

text: such

text: lator

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text: states

text: shown

text: in

text: Fig.

text: It

text: is

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text: as

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text: entanglement.

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text: if

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text: of

text: such

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text: 4 Conclusion In this paper, we propose a new collaborative practice-based naming system and mobile devices to provide real-time support for the recovery. Through it, we can not only relates to stroke survivors, their caregivers, therapists and doctors play an important role in the rehabilitation process. As a feedback capacitor-based mobile application, called armstrokes, we proposed to provide rehabilitation through interactive games limb rehabilitation. Users can get real-time stroke of visual, audio and tactile feed- back about their practice results, and see their performance. Caregivers, therapists and doctors can immediately monitor the rehabilitation process and practice outcomes for patients through the system, and then follow the patient through treatment such as reminders, encouragement and warning. References [1] Shaughnessy, M., and Resnick, B., 2006. “Testing a model of post-stroke exercise behavior”. In Rehabilitation Nursing: The Ofﬁcial Journal of the Assoc. of Rehabilitation, pp. 15–31. [2] Golomb, M., McDonald, B., and Warden, S. J., 2013. “Virtual rehabilitation with video games: A new frontier for occupational therapy”. Occupational Therapy Computer Graphics, 40(10), pp. 613–624. [3] Song, H., Xu, X., and Shi, K., 2010. “Television promoting of in-home rehabilitation exercises: Promising results.”. In User Modeling, Adaptation and Personalization, 15(5), November, pp. 745– 770. [4] Liu, X.-M., and Sun, J.-G., 2014. “Projecting points onto planar parametric curves by local biarc approximation”. Paciﬁc Graphics Short Papers. [5] Kison, W. S., 2002. “Computer vision assisted virtual reality calibration”. IEEE Transactions on Robotics and Automation, 15(3), pp. 450–464. [6] Wright, W. S., and Chalfant, E. C., 1997. “Computer vision assisted semi-automatic virtual reality calibration”. In Robotics and Automation, 1997. Proceedings., 1997 IEEE International Confer- ence on, Vol. 2. [7] Shen, J., Su, P. C., c. S. Cheung, S., and Zhao, J., 2013. “Virtual mirror rendering with station- ary rgb-d cameras and stored 3-d background”. IEEE Transactions on Image Processing, 22(9), pp. 3433–3448. [8] DiVerdi, S., and Hollerer, T., 2008. “Groundcam: A tracking modality for mobile mixed reality”.

label: section

text: Conclusion

text: In

text: this

text: paper,

text: we

text: propose

text: a

text: new

text: collaborative

text: practice-based

text: naming

text: system

text: and

text: mobile

text: devices

text: to

text: provide

text: real-time

text: support

text: for

text: the

text: recovery.

text: Through

text: it,

text: we

text: can

text: not

text: only

text: relates

text: to

text: stroke

text: survivors,

text: their

text: caregivers,

text: therapists

text: and

text: doctors

text: play

text: an

text: important

text: role

text: in

text: the

text: rehabilitation

text: process.

text: As

text: a

text: feedback

text: capacitor-based

text: mobile

text: application,

text: called

text: armstrokes,

text: we

text: proposed

text: to

text: provide

text: rehabilitation

text: through

text: interactive

text: games

text: limb

text: rehabilitation.

text: Users

text: can

text: get

text: real-time

text: stroke

text: of

text: visual,

text: audio

text: and

text: tactile

text: feed-

text: back

text: about

text: their

text: practice

text: results,

text: and

text: see

text: their

text: performance.

text: Caregivers,

text: therapists

text: and

text: doctors

text: can

text: immediately

text: monitor

text: the

text: rehabilitation

text: process

text: and

text: practice

text: outcomes

text: for

text: patients

text: through

text: the

text: system,

text: and

text: then

text: follow

text: the

text: patient

text: through

text: treatment

text: such

text: as

text: reminders,

text: encouragement

text: and

text: warning.

text: References

text: Shaughnessy,

text: M.,

text: and

text: Resnick,

text: B.,

text: “Testing

text: a

text: model

text: of

text: post-stroke

text: exercise

text: behavior”.

text: In

text: Rehabilitation

text: Nursing:

text: The

text: Ofﬁcial

text: Journal

text: of

text: the

text: Assoc.

text: of

text: Rehabilitation,

text: pp.

text: Golomb,

text: M.,

text: McDonald,

text: B.,

text: and

text: Warden,

text: S.

text: J.,

text: “Virtual

text: rehabilitation

text: with

text: video

text: games:

text: A

text: new

text: frontier

text: for

text: occupational

text: therapy”.

text: Occupational

text: Therapy

text: Computer

text: Graphics,

text: pp.

text: Song,

text: H.,

text: Xu,

text: X.,

text: and

text: Shi,

text: K.,

text: “Television

text: promoting

text: of

text: in-home

text: rehabilitation

text: exercises:

text: Promising

text: results.”.

text: In

text: User

text: Modeling,

text: Adaptation

text: and

text: Personalization,

text: November,

text: pp.

text: Liu,

text: X.-M.,

text: and

text: Sun,

text: J.-G.,

text: “Projecting

text: points

text: onto

text: planar

text: parametric

text: curves

text: by

text: local

text: biarc

text: approximation”.

text: Paciﬁc

text: Graphics

text: Short

text: Papers.

text: Kison,

text: W.

text: S.,

text: “Computer

text: vision

text: assisted

text: virtual

text: reality

text: calibration”.

text: IEEE

text: Transactions

text: on

text: Robotics

text: and

text: Automation,

text: pp.

text: Wright,

text: W.

text: S.,

text: and

text: Chalfant,

text: E.

text: C.,

text: “Computer

text: vision

text: assisted

text: semi-automatic

text: virtual

text: reality

text: calibration”.

text: In

text: Robotics

text: and

text: Automation,

text: Proceedings.,

text: IEEE

text: International

text: Confer-

text: ence

text: on,

text: Vol.

text: Shen,

text: J.,

text: Su,

text: P.

text: C.,

text: c.

text: S.

text: Cheung,

text: S.,

text: and

text: Zhao,

text: J.,

text: “Virtual

text: mirror

text: rendering

text: with

text: station-

text: ary

text: rgb-d

text: cameras

text: and

text: stored

text: background”.

text: IEEE

text: Transactions

text: on

text: Image

text: Processing,

text: pp.

text: DiVerdi,

text: S.,

text: and

text: Hollerer,

text: T.,

text: “Groundcam:

text: A

text: tracking

text: modality

text: for

text: mobile

text: mixed

text: reality”.


text: the results in Theorem 2 can still hold for this choice of λ by resorting to Lemma 6 and 0 similar arguments in the proof of Theorem 2. 3.3 Veriﬁcation of Condition 4 Since Condition 4 is a key assumption for proving Theorem 2, we provide some suf- (cid:101) (cid:101) (cid:101) (cid:101) ﬁcient conditions that ensures this assumption on the augmented random design matrix (cid:101) · · · X = (x , , x ). Denote by Σ the population covariance matrix of the augmented covari- 1 p − ate vector consisting of p main eﬀects X ’s and p(p 1)/2 interactions X X ’s. (cid:101) j k (cid:96) ∈ R Condition 6. There exists some constant K > 0 such that for δ = (δT , δT )T p, 1 2 (cid:16) (cid:17) (cid:101) (cid:101) (cid:101) ≥ (cid:107) (cid:107) ∨ (cid:107) (cid:107) ≥ min δT Σδ K and min δT Σδ/ δ δ K, (cid:107) (cid:107) (cid:107) (cid:107) (cid:54) (cid:107) (cid:107) ≤ (cid:107) (cid:107) 1 2 2 2 δ 2=1, δ 0<2s δ=0, δ2 1 7 δ1 1 (cid:101) ∈ R where δ s and δ is a subvector of δ consisting of the s largest components in 1 2 2 magnitude. (cid:101) Condition 6 is satisﬁed if the smallest eigenvalue of Σ is assumed to be bounded away from zero. Such a condition is in fact much weaker than the minimum eigenvalue assumption, since it is the population version of a mild sparse eigenvalue assumption and the restricted (cid:101) eigenvalue assumption. The following theorem shows that under some mild assumptions, (cid:101) Condition 4 holds for the full augmented design matrix X and thus holds naturally for any × ≤ n q sub-design matrix with q p and the sure screening property. Theorem 3. Assume that Condition 6 holds, there exist some constants α , c > 0 such 1 1 | | ≤ − − that for any t > 0, P ( Xj > t) c1 exp( c1 1tα1) for each j, s = O(nξ0), and log p = { }− ≤ { } o(nmin α1/4,1 2ξ0) with constant 0 ξ0 < min α1/8, 1/2 . Then Condition 4 holds with { }− − nmin α1/4,1 2ξ0 = O( log an). 4 Numerical studies In this section, we design two simulation examples to verify the theoretical results and 17

text: the

text: results

text: in

text: Theorem

text: can

text: still

text: hold

text: for

text: this

text: choice

text: of

text: λ

text: by

text: resorting

text: to

text: Lemma

text: and

text: similar

text: arguments

text: in

text: the

text: proof

text: of

text: Theorem

text: Veriﬁcation

text: of

text: Condition

text: Since

text: Condition

text: is

text: a

text: key

text: assumption

text: for

text: proving

text: Theorem

text: we

text: provide

text: some

text: suf-

text: ﬁcient

text: conditions

text: that

text: ensures

text: this

text: assumption

text: on

text: the

text: augmented

text: random

text: design

text: matrix

text: X

text: (x

text: x

text: Denote

text: by

text: Σ

text: the

text: population

text: covariance

text: matrix

text: of

text: the

text: augmented

text: covari-

text: p

text: ate

text: vector

text: consisting

text: of

text: p

text: main

text: eﬀects

text: X

text: ’s

text: and

text: p(p

text: interactions

text: X

text: X

text: ’s.

text: j

text: k

text: R

text: Condition

text: There

text: exists

text: some

text: constant

text: K

text: such

text: that

text: for

text: δ

text: (δT

text: δT

text: )T

text: p,

text: min

text: δT

text: Σδ

text: K

text: and

text: min

text: δT

text: Σδ/

text: δ

text: δ

text: K,

text: δ

text: δ

text: δ2

text: δ1

text: R

text: where

text: δ

text: s

text: and

text: δ

text: is

text: a

text: subvector

text: of

text: δ

text: consisting

text: of

text: the

text: s

text: largest

text: components

text: in

text: magnitude.

text: Condition

text: is

text: satisﬁed

text: if

text: the

text: smallest

text: eigenvalue

text: of

text: Σ

text: is

text: assumed

text: to

text: be

text: bounded

text: away

text: from

text: zero.

text: Such

text: a

text: condition

text: is

text: in

text: fact

text: much

text: weaker

text: than

text: the

text: minimum

text: eigenvalue

text: assumption,

text: since

text: it

text: is

text: the

text: population

text: version

text: of

text: a

text: mild

text: sparse

text: eigenvalue

text: assumption

text: and

text: the

text: restricted

text: eigenvalue

text: assumption.

text: The

text: following

text: theorem

text: shows

text: that

text: under

text: some

text: mild

text: assumptions,

text: Condition

text: holds

text: for

text: the

text: full

text: augmented

text: design

text: matrix

text: X

text: and

text: thus

text: holds

text: naturally

text: for

text: any

text: n

text: q

text: sub-design

text: matrix

text: with

text: q

text: p

text: and

text: the

text: sure

text: screening

text: property.

text: Theorem

text: Assume

text: that

text: Condition

text: holds,

text: there

text: exist

text: some

text: constants

text: α

text: c

text: such

text: that

text: for

text: any

text: t

text: P

text: Xj

text: t)

text: exp(

text: for

text: each

text: j,

text: s

text: and

text: log

text: p

text: o(nmin

text: with

text: constant

text: ξ0

text: min

text: Then

text: Condition

text: holds

text: with

text: nmin

text: O(

text: log

text: an).

text: Numerical

text: studies

text: In

text: this

text: section,

text: we

text: design

text: two

text: simulation

text: examples

text: to

text: verify

text: the

text: theoretical

text: results

text: and


text: examine the ﬁnite-sample performance of the suggested approach IP. We also present an analysis of a prostate cancer data set. 4.1 Feature screening performance We start with comparing IP with several recent feature screening procedures: the SIS, DC-SIS (Li et al., 2012), and SIRI (Jiang and Liu, 2014). The SIRI is an iterative pro- cedure that alternates between a large-scale variable screening step and a moderate-scale variable selection step when the dimensionality p is large. Since all other screening methods are non-iterative, in this section, we compare the initial screening step of SIRI with other methods and name the screening only procedure as SIRI*. The full iterative SIRI will be included in Section later for comparison of variable selection. SIRI*, SIS, and DC-SIS each return a set of variables without distinguishing between important main eﬀects and active interaction variables. Thus, for each method, we construct interactions using all possible pairwise interactions of the recruited variables. By doing so, the strong heredity assumption is enforced. We name the resulting procedures as SIRI*2, SIS2, and DC-SIS2 to distinguish them from their original versions. (cid:98) (cid:98) (cid:99) (cid:98) (cid:98) For IP, as mentioned in Section , we retain the top [n/(log n)] variables in each of sets A B (cid:98) M A ∪ B and deﬁned in (5) and (7), respectively. The features in the union set = are A used as main eﬀects while variables in set are used to build interactions in the selection (cid:99) step of IP. To ensure a fair comparison, the numbers of variables kept in SIRI*2, SIS2, and M DC-SIS2 are all equal to the cardinality of , which is up to 2[n/(log n)]. Example 1 (Gaussian distribution). We consider the following four interaction models linking the covariates X ’s to the response Y : j • M1 (strong heredity): Y = 2X + 2X + 3X X + ε , 1 5 1 5 1 • M2 (weak heredity): Y = 2X + 2X + 3X X + ε , 1 10 1 5 2 • M3 (anti-heredity): Y = 2X + 2X + 3X X + ε , 10 15 1 5 3 • M4 (interactions only): Y = 3X X + 3X X + ε , 1 5 10 15 4 · · · ∼ | − | where the covariate vector x = (X , , X )T N(0, Σ) with Σ = (ρ j k ) ≤ ≤ and the 1 p 1 j,k p ∼ ∼ ∼ ∼ errors ε N(0, 2.52), ε N(0, 22), ε N(0, 22), and ε N(0, 1.52) are independent of 1 2 3 4 x. The ﬁrst two models M1 and M2 satisfy the heredity assumption (either strong or weak), 18

text: examine

text: the

text: ﬁnite-sample

text: performance

text: of

text: the

text: suggested

text: approach

text: IP.

text: We

text: also

text: present

text: an

text: analysis

text: of

text: a

text: prostate

text: cancer

text: data

text: set.

text: Feature

text: screening

text: performance

text: We

text: start

text: with

text: comparing

text: IP

text: with

text: several

text: recent

text: feature

text: screening

text: procedures:

text: the

text: SIS,

text: DC-SIS

text: (Li

text: et

text: al.,

text: and

text: SIRI

text: (Jiang

text: and

text: Liu,

text: The

text: SIRI

text: is

text: an

text: iterative

text: pro-

text: cedure

text: that

text: alternates

text: between

text: a

text: large-scale

text: variable

text: screening

text: step

text: and

text: a

text: moderate-scale

text: variable

text: selection

text: step

text: when

text: the

text: dimensionality

text: p

text: is

text: large.

text: Since

text: all

text: other

text: screening

text: methods

text: are

text: non-iterative,

text: in

text: this

text: section,

text: we

text: compare

text: the

text: initial

text: screening

text: step

text: of

text: SIRI

text: with

text: other

text: methods

text: and

text: name

text: the

text: screening

text: only

text: procedure

text: as

text: SIRI*.

text: The

text: full

text: iterative

text: SIRI

text: will

text: be

text: included

text: in

text: Section

text: later

text: for

text: comparison

text: of

text: variable

text: selection.

text: SIRI*,

text: SIS,

text: and

text: DC-SIS

text: each

text: return

text: a

text: set

text: of

text: variables

text: without

text: distinguishing

text: between

text: important

text: main

text: eﬀects

text: and

text: active

text: interaction

text: variables.

text: Thus,

text: for

text: each

text: method,

text: we

text: construct

text: interactions

text: using

text: all

text: possible

text: pairwise

text: interactions

text: of

text: the

text: recruited

text: variables.

text: By

text: doing

text: so,

text: the

text: strong

text: heredity

text: assumption

text: is

text: enforced.

text: We

text: name

text: the

text: resulting

text: procedures

text: as

text: and

text: to

text: distinguish

text: them

text: from

text: their

text: original

text: versions.

text: For

text: IP,

text: as

text: mentioned

text: in

text: Section

text: we

text: retain

text: the

text: top

text: [n/(log

text: n)]

text: variables

text: in

text: each

text: of

text: sets

text: A

text: B

text: M

text: A

text: B

text: and

text: deﬁned

text: in

text: and

text: respectively.

text: The

text: features

text: in

text: the

text: union

text: set

text: are

text: A

text: used

text: as

text: main

text: eﬀects

text: while

text: variables

text: in

text: set

text: are

text: used

text: to

text: build

text: interactions

text: in

text: the

text: selection

text: step

text: of

text: IP.

text: To

text: ensure

text: a

text: fair

text: comparison,

text: the

text: numbers

text: of

text: variables

text: kept

text: in

text: and

text: M

text: are

text: all

text: equal

text: to

text: the

text: cardinality

text: of

text: which

text: is

text: up

text: to

text: n)].

text: Example

text: (Gaussian

text: distribution).

text: We

text: consider

text: the

text: following

text: four

text: interaction

text: models

text: linking

text: the

text: covariates

text: X

text: ’s

text: to

text: the

text: response

text: Y

text: j

text: (strong

text: heredity):

text: Y

text: X

text: ε

text: (weak

text: heredity):

text: Y

text: X

text: ε

text: (anti-heredity):

text: Y

text: X

text: ε

text: (interactions

text: only):

text: Y

text: X

text: X

text: ε

text: where

text: the

text: covariate

text: vector

text: x

text: (X

text: X

text: )T

text: Σ)

text: with

text: Σ

text: (ρ

text: j

text: k

text: and

text: the

text: p

text: j,k

text: p

text: errors

text: ε

text: ε

text: ε

text: and

text: ε

text: are

text: independent

text: of

text: x.

text: The

text: ﬁrst

text: two

text: models

text: and

text: satisfy

text: the

text: heredity

text: assumption

text: (either

text: strong

text: or

text: weak),


text: Holt et al. (2010) claimed Hepsin (HPN) as one of the most consistently overexpressed genes in prostate cancer. In addition, evidence of the association between TARP gene variants and prostate cancer risk has been shown in Wolfgang et al. (2000), Oh et al. (2004), and Hillerdal et al. (2012). Note that the gene ERG was missed by both SIS2-Enet and DC-SIS2-Enet in the top 10 main eﬀects, but it was selected by IP-Enet as a main eﬀect and part of an × interaction (SLC7A1 ERG). There are a wide range of studies investigating the eﬀect of ERG on prostate cancer (Furusato et al., 2010; Klezovitch et al., 2008). × The most frequently selected interaction DPT S100A4 by SIS2-Enet and DC-SIS2-Enet × is also among the top 10 list by IP-Enet. Two more interactions, RARRES2 KLK3 and × MAF NELL2, are also among the top 10 lists by both IP-Enet and SIS2-Enet. However, × × some interactions involving PRKDC (PRKDC CFD and PRKDC KLK3) were very often selected by IP-Enet but missed by the other two methods. There are studies showing that PRKDC is associated with prostate cancer (McCarthy et al., 2013). Such a ﬁnding favors × × the results of IP that the interactions PRKDC CFD and PRKDC KLK3 were identiﬁed to be associated with the phenotype. 5 Discussion We have considered in this paper the problem of interaction identiﬁcation in ultra-high dimensions. The proposed method IP based on a new interaction screening procedure and post-screening variable selection is computationally eﬃcient, and capable of reducing dimen- sionality from a large scale to a moderate one and recovering important interactions and main eﬀects. To simplify the technical presentation, our analysis has been focused on the linear pairwise interaction models. Screening for main eﬀects in more general model settings has been explored by many researchers; see, for example, Fan and Song (2010), Fan et al. (2011), Chang et al. (2013), and Cheng et al. (2014). It would be interesting to extend the interaction screening idea of IP to these and other more general model frameworks such as the generalized linear models, nonparametric models, and survival models with interactions. The key idea of IP is to use diﬀerent marginal utilities to screen interactions and main eﬀects separately. As such, it can suﬀer from the same potential issues as the SIS. First, some noise interactions or main eﬀects that are highly correlated with the important ones 23

text: Holt

text: et

text: al.

text: claimed

text: Hepsin

text: (HPN)

text: as

text: one

text: of

text: the

text: most

text: consistently

text: overexpressed

text: genes

text: in

text: prostate

text: cancer.

text: In

text: addition,

text: evidence

text: of

text: the

text: association

text: between

text: TARP

text: gene

text: variants

text: and

text: prostate

text: cancer

text: risk

text: has

text: been

text: shown

text: in

text: Wolfgang

text: et

text: al.

text: Oh

text: et

text: al.

text: and

text: Hillerdal

text: et

text: al.

text: Note

text: that

text: the

text: gene

text: ERG

text: was

text: missed

text: by

text: both

text: and

text: in

text: the

text: top

text: main

text: eﬀects,

text: but

text: it

text: was

text: selected

text: by

text: IP-Enet

text: as

text: a

text: main

text: eﬀect

text: and

text: part

text: of

text: an

text: interaction

text: ERG).

text: There

text: are

text: a

text: wide

text: range

text: of

text: studies

text: investigating

text: the

text: eﬀect

text: of

text: ERG

text: on

text: prostate

text: cancer

text: (Furusato

text: et

text: al.,

text: Klezovitch

text: et

text: al.,

text: The

text: most

text: frequently

text: selected

text: interaction

text: DPT

text: by

text: and

text: is

text: also

text: among

text: the

text: top

text: list

text: by

text: IP-Enet.

text: Two

text: more

text: interactions,

text: and

text: MAF

text: are

text: also

text: among

text: the

text: top

text: lists

text: by

text: both

text: IP-Enet

text: and

text: However,

text: some

text: interactions

text: involving

text: PRKDC

text: (PRKDC

text: CFD

text: and

text: PRKDC

text: were

text: very

text: often

text: selected

text: by

text: IP-Enet

text: but

text: missed

text: by

text: the

text: other

text: two

text: methods.

text: There

text: are

text: studies

text: showing

text: that

text: PRKDC

text: is

text: associated

text: with

text: prostate

text: cancer

text: (McCarthy

text: et

text: al.,

text: Such

text: a

text: ﬁnding

text: favors

text: the

text: results

text: of

text: IP

text: that

text: the

text: interactions

text: PRKDC

text: CFD

text: and

text: PRKDC

text: were

text: identiﬁed

text: to

text: be

text: associated

text: with

text: the

text: phenotype.

text: Discussion

text: We

text: have

text: considered

text: in

text: this

text: paper

text: the

text: problem

text: of

text: interaction

text: identiﬁcation

text: in

text: ultra-high

text: dimensions.

text: The

text: proposed

text: method

text: IP

text: based

text: on

text: a

text: new

text: interaction

text: screening

text: procedure

text: and

text: post-screening

text: variable

text: selection

text: is

text: computationally

text: eﬃcient,

text: and

text: capable

text: of

text: reducing

text: dimen-

text: sionality

text: from

text: a

text: large

text: scale

text: to

text: a

text: moderate

text: one

text: and

text: recovering

text: important

text: interactions

text: and

text: main

text: eﬀects.

text: To

text: simplify

text: the

text: technical

text: presentation,

text: our

text: analysis

text: has

text: been

text: focused

text: on

text: the

text: linear

text: pairwise

text: interaction

text: models.

text: Screening

text: for

text: main

text: eﬀects

text: in

text: more

text: general

text: model

text: settings

text: has

text: been

text: explored

text: by

text: many

text: researchers;

text: see,

text: for

text: example,

text: Fan

text: and

text: Song

text: Fan

text: et

text: al.

text: Chang

text: et

text: al.

text: and

text: Cheng

text: et

text: al.

text: It

text: would

text: be

text: interesting

text: to

text: extend

text: the

text: interaction

text: screening

text: idea

text: of

text: IP

text: to

text: these

text: and

text: other

text: more

text: general

text: model

text: frameworks

text: such

text: as

text: the

text: generalized

text: linear

text: models,

text: nonparametric

text: models,

text: and

text: survival

text: models

text: with

text: interactions.

text: The

text: key

text: idea

text: of

text: IP

text: is

text: to

text: use

text: diﬀerent

text: marginal

text: utilities

text: to

text: screen

text: interactions

text: and

text: main

text: eﬀects

text: separately.

text: As

text: such,

text: it

text: can

text: suﬀer

text: from

text: the

text: same

text: potential

text: issues

text: as

text: the

text: SIS.

text: First,

text: some

text: noise

text: interactions

text: or

text: main

text: eﬀects

text: that

text: are

text: highly

text: correlated

text: with

text: the

text: important

text: ones


text: Cordell, H. J. (2009), “Detecting gene–gene interactions that underlie human diseases,” Nature Reviews Genetics, 10, 392–404. Cui, H., Li, R. and Zhong, W. (2015), “Model-free feature screening for ultrahigh dimensional discriminant analysis,” Journal of the American Statistical Association, 110, 630–641. Culverhouse, R., Suarez, B. K., Lin, J. and Reich, T. (2002), “A perspective on epistasis: limits of models displaying no main eﬀect,” The American Journal of Human Genetics, 70, 461–471. Fan, J. and Fan, Y. (2008), “High dimensional classiﬁcation using features annealed inde- pendence rules,” The Annals of Statistics, 36, 2605–2637. Fan, J., Feng, Y. and Song, R. (2011), “Nonparametric independence screening in sparse ultra-high-dimensional additive models,” Journal of the American Statistical Association, 106, 544–557. Fan, J. and Li, R. (2001), “Variable selection via nonconcave penalized likelihood and its oracle properties,” Journal of the American Statistical Association, 96, 1348–1360. Fan, J. and Lv, J. (2008), “Sure independence screening for ultrahigh dimensional feature space” (with discussion), Journal of the Royal Statistical Society, Series B, 70, 849–911. Fan, J., Samworth, R. and Wu, Y. (2009), “Ultrahigh dimensional feature selection: beyond the linear model,” Journal of Machine Learning Research, 10, 2013–2038. Fan, J. and Song, R. (2010), “Sure independence screening in generalized linear models with NP-dimensionality,” The Annals of Statistics, 38, 3567–3604. Fan, Y. and Lv, J. (2014), “Asymptotic properties for combined L and concave regulariza- 1 tion,” Biometrika, 101, 57–70. Furusato, B., Tan, S. H., Young, D., Dobi, A., Sun, C., Mohamed, A. A., Thangapazham, R., Chen, Y., McMaster, G., Sreenath, T., Petrovics, G., McLeod, D. G., Srivastava, S. and Sesterhenn, I. A. (2010), “ERG oncoprotein expression in prostate cancer: clonal pro- gression of ERG-positive tumor cells and potential for ERG-based stratiﬁcation,” Prostate Cancer and Prostatic Diseases, 13, 228–237. 26

text: Cordell,

label: reference

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text: Genetics,

text: Cui,

text: H.,

text: Li,

text: R.

text: and

text: Zhong,

text: W.

text: “Model-free

text: feature

text: screening

text: for

text: ultrahigh

text: dimensional

text: discriminant

text: analysis,”

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Culverhouse,

text: R.,

text: Suarez,

text: B.

text: K.,

text: Lin,

text: J.

text: and

text: Reich,

text: T.

text: “A

text: perspective

text: on

text: epistasis:

text: limits

text: of

text: models

text: displaying

text: no

text: main

text: eﬀect,”

text: The

text: American

text: Journal

text: of

text: Human

text: Genetics,

text: Fan,

text: J.

text: and

text: Fan,

text: Y.

text: “High

text: dimensional

text: classiﬁcation

text: using

text: features

text: annealed

text: inde-

text: pendence

text: rules,”

text: The

text: Annals

text: of

text: Statistics,

text: Fan,

text: J.,

text: Feng,

text: Y.

text: and

text: Song,

text: R.

text: “Nonparametric

text: independence

text: screening

text: in

text: sparse

text: ultra-high-dimensional

text: additive

text: models,”

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Fan,

text: J.

text: and

text: Li,

text: R.

text: “Variable

text: selection

text: via

text: nonconcave

text: penalized

text: likelihood

text: and

text: its

text: oracle

text: properties,”

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Fan,

text: J.

text: and

text: Lv,

text: J.

text: “Sure

text: independence

text: screening

text: for

text: ultrahigh

text: dimensional

text: feature

text: space”

text: (with

text: discussion),

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society,

text: Series

text: B,

text: Fan,

text: J.,

text: Samworth,

text: R.

text: and

text: Wu,

text: Y.

text: “Ultrahigh

text: dimensional

text: feature

text: selection:

text: beyond

text: the

text: linear

text: model,”

text: Journal

text: of

text: Machine

text: Learning

text: Research,

text: Fan,

text: J.

text: and

text: Song,

text: R.

text: “Sure

text: independence

text: screening

text: in

text: generalized

text: linear

text: models

text: with

text: NP-dimensionality,”

text: The

text: Annals

text: of

text: Statistics,

text: Fan,

text: Y.

text: and

text: Lv,

text: J.

text: “Asymptotic

text: properties

text: for

text: combined

text: L

text: and

text: concave

text: regulariza-

text: tion,”

text: Biometrika,

text: Furusato,

text: B.,

text: Tan,

text: S.

text: H.,

text: Young,

text: D.,

text: Dobi,

text: A.,

text: Sun,

text: C.,

text: Mohamed,

text: A.

text: A.,

text: Thangapazham,

text: R.,

text: Chen,

text: Y.,

text: McMaster,

text: G.,

text: Sreenath,

text: T.,

text: Petrovics,

text: G.,

text: McLeod,

text: D.

text: G.,

text: Srivastava,

text: S.

text: and

text: Sesterhenn,

text: I.

text: A.

text: “ERG

text: oncoprotein

text: expression

text: in

text: prostate

text: cancer:

text: clonal

text: pro-

text: gression

text: of

text: ERG-positive

text: tumor

text: cells

text: and

text: potential

text: for

text: ERG-based

text: stratiﬁcation,”

text: Prostate

text: Cancer

text: and

text: Prostatic

text: Diseases,


text: Table 6: The means and standard errors (in parentheses) of classiﬁcation errors and median model sizes in prostate cancer data analysis. Method Classiﬁcation error Median model size SIS2-Enet 0.0754 (0.0030) 75 DC-SIS2-Enet 0.0745 (0.0031) 70 IP-Enet 0.0681 (0.0033) 106 34 ##LTLine## ##LTLine## ##LTLine##

text: Table

text: The

label: caption

text: means

text: and

text: standard

text: errors

text: (in

text: parentheses)

text: of

text: classiﬁcation

text: errors

text: and

text: median

text: model

text: sizes

text: in

text: prostate

text: cancer

text: data

text: analysis.

text: Method

text: Classiﬁcation

text: error

text: Median

text: model

text: size

text: IP-Enet

text: ##LTLine##

font: default

text: ##LTLine##

text: ##LTLine##


text:       2γ2, j = 1 or s + 1    ≥ cov(X2, J2) = 2γ2(σ2 + σ2 ) = 2γ2ρ2, j = 2 or s or s + 2 when s 3. j 2 j1 j,s+1 ≤ ≤ − ≥ 0, 3 j s 1 or j s + 3 ≥ ∈ A { } So it holds that cov(X2, Y 2) = 0 for all j s+3, and cov(X2, Y 2) > 0 for j = 1, s+1 . j j A { } Case 3: = s + 1, s + 2 . Then J = γX X and cov(X2, J2) = 2γ2(σ2 σ + 2 s s+1 j 2 js s+1,s+1 4σ σ σ + σ2 σ ). Thus js j,s+1 s,s+1 j,s+1 ss      2γ2(1 + 5ρ2), j = s or s + 1,    − cov(X2, J2) = 2γ2ρ2, j = s 1 or s + 2, j 2 0, otherwise. ≥ ∈ A So we have that cov(X2, Y 2) = 0 for all j s + 3, and cov(X2, Y 2) > 0 for j = j j { } s + 1, s + 2 . ∈ A ≥ { Therefore, cov(X2, Y 2) > 0 for all j , whereas cov(X2, Y 2) = 0 for all j max s + j (cid:113) j } ≥ 2, 4 for Case 1, and cov(X2, Y 2) = 0 for all j s + 3 for Cases 2 and 3. Note that j corr(X2, Y 2) = cov(X2, Y 2)/ var(X2)var(Y 2). This ensures that the correlations between j j j X2 and Y 2 are nonzero for those active interaction variables. In other words, using corr(X2, Y 2) j j as the marginal utility can still single out active interaction variables. Appendix D: Proofs of Proposition 1 and Theorems 1–3 (cid:80) (cid:80) (cid:80) D.1. Proof of Proposition 1 − Let J = p β X and J = p 1 p γ X X . Then our interaction model (1) 1 j=1 j j 2 k=1 (cid:96)=k+1 k(cid:96) k (cid:96) ∈ { · · · } can be written as Y = β + J + J + ε. For each j 1, , p , the covariance between X2 0 1 2 j and Y 2 can be expressed as cov(X2, Y 2) = cov(X2, J2) + cov(X2, J2) + cov(X2, ε2) + 2β cov(X2, J ) j j 1 j 2 j 0 j 1 + 2β cov(X2, J ) + 2β cov(X2, ε) + 2cov(X2, J J ) 0 j 2 0 j j 1 2 + 2cov(X2, J ε) + 2cov(X2, J ε). (D.1) j 1 j 2 8 ##LTLine## ##LTLine##

text: j

text: or

text: s

text: σ2

text: j

text: or

text: s

text: or

text: s

text: when

text: s

text: j

text: j

text: s

text: or

text: j

text: s

text: A

text: So

text: it

text: holds

text: that

text: Y

text: for

text: all

text: j

text: and

text: Y

text: for

text: j

text: j

text: j

text: A

text: Case

text: s

text: s

text: Then

text: J

text: γX

text: X

text: and

text: σ

text: s

text: j

text: js

text: 4σ

text: σ

text: σ

text: σ2

text: σ

text: Thus

text: js

text: ss

text: j

text: s

text: or

text: s

text: j

text: s

text: or

text: s

text: j

text: otherwise.

text: A

text: So

text: we

text: have

text: that

text: Y

text: for

text: all

text: j

text: s

text: and

text: Y

text: for

text: j

text: j

text: j

text: s

text: s

text: A

text: Therefore,

text: Y

text: for

text: all

text: j

text: whereas

text: Y

text: for

text: all

text: j

text: max

text: s

text: j

text: j

text: for

text: Case

text: and

text: Y

text: for

text: all

text: j

text: s

text: for

text: Cases

text: and

text: Note

text: that

text: j

text: Y

text: Y

text: This

text: ensures

text: that

text: the

text: correlations

text: between

text: j

text: j

text: j

text: and

text: Y

text: are

text: nonzero

text: for

text: those

text: active

text: interaction

text: variables.

text: In

text: other

text: words,

text: using

text: Y

text: j

text: j

text: as

text: the

text: marginal

text: utility

text: can

text: still

text: single

text: out

text: active

text: interaction

text: variables.

text: Appendix

text: D:

text: Proofs

text: of

text: Proposition

text: and

text: Theorems

text: Proof

text: of

text: Proposition

text: Let

text: J

text: p

text: β

text: X

text: and

text: J

text: p

text: p

text: γ

text: X

text: X

text: Then

text: our

text: interaction

text: model

text: j

text: j

text: k

text: can

text: be

text: written

text: as

text: Y

text: β

text: J

text: J

text: ε.

text: For

text: each

text: j

text: p

text: the

text: covariance

text: between

text: j

text: and

text: Y

text: can

text: be

text: expressed

text: as

text: Y

text: 2β

text: J

text: j

text: j

text: j

text: j

text: j

text: 2β

text: J

text: 2β

text: ε)

text: J

text: J

text: j

text: j

text: j

text: J

text: ε)

text: J

text: ε).

text: j

text: j

text: ##LTLine##

text: ##LTLine##


text: (cid:40)(cid:34) (cid:35) (cid:41) we deduce (cid:2) (cid:3) (cid:88) 1/2 E(|X1j|(cid:107)x1,B(cid:107)1IΦc) ≤ (cid:40)E(X12j(cid:107)x1,B(cid:107)21)P (Φc1) 1/2 ≤ (cid:41) s2  E(X12jX12(cid:96)) P (Φc1)  1 (cid:88) ∈B (cid:88) (cid:96)  1/2 1/2 ≤ − | | 2 1s [E(X4 ) + E(X4 )] P ( X > M ) (cid:101) 2 ∈B 1j 1(cid:96) ∈M∪{ } i(cid:96) 6 (cid:96) (cid:96) j ≤ − Cs (1 + s + 2s )1/2 exp[ Mα1/(2c )] 2 2 1 6 1 (cid:101) (cid:101) for some positive constant C, where the last inequality follows from Condition 2 and Lemma | |(cid:107) (cid:107) I ≤ − 2. Similarly, we have E( X1j z1,I 1 Φc1) Cs1(1 + s2 + 2s1)1/2 exp[ M6α1/(2c1)]. This together with the above inequalities entails that (cid:101) | | ≤ − E( T ) C C(s + s )(1 + s + 2s )1/2 exp[ Mα1/(2c )]. j2 0 1 2 2 1 6 1 If we choose M6 = nη6 with η6 > 0, then by Condition 1, for any positive constant C, when n is suﬃciently large, (cid:101) | | ≤ − − E( T ) C C(nξ1 + nξ2)(1 + nξ2 + 2nξ1)1/2 exp[ nα1η6/(2c )] < Cn κ2/6 (D.44) j2 0 1 ≤ ≤ holds uniformly for all 1 j p. The above inequality together with (D.42) ensures that | − | ≥ − P ( max S E(S ) Cn κ2/2) ≤ ≤ j1 j1 1 j p ≤ | − | ≥ − | | ≥ − P ( max T E(T ) Cn κ2/6) + P ( max T Cn κ2/6) (D.45) ≤ ≤ j1 j1 ≤ ≤ j2 1 j p 1 j p for all n is suﬃciently large. Thus we only need to establish the probability bound for each term on the right hand side of (D.45). | − | First consider max ≤ ≤ T E(T ) . Using similar arguments as for proving (D.43), 1 j p j1 j1 26

text: we

text: deduce

text: P

text: ∈B

text: P

text: X

text: M

text: ∈B

text: ∈M∪{

text: j

text: Cs

text: s

text: exp[

text: for

text: some

text: positive

text: constant

text: C,

text: where

text: the

text: last

text: inequality

text: follows

text: from

text: Condition

text: and

text: Lemma

text: I

text: Similarly,

text: we

text: have

text: E(

text: exp[

text: This

text: together

text: with

text: the

text: above

text: inequalities

text: entails

text: that

text: E(

text: T

text: C

text: C(s

text: s

text: s

text: exp[

text: If

text: we

text: choose

text: with

text: η6

text: then

text: by

text: Condition

text: for

text: any

text: positive

text: constant

text: C,

text: when

text: n

text: is

text: suﬃciently

text: large,

text: E(

text: T

text: C

text: exp[

text: Cn

text: holds

text: uniformly

text: for

text: all

text: j

text: p.

text: The

text: above

text: inequality

text: together

text: with

text: ensures

text: that

text: P

text: max

text: S

text: E(S

text: Cn

text: j

text: p

text: P

text: max

text: T

text: E(T

text: Cn

text: P

text: max

text: T

text: Cn

text: j

text: p

text: j

text: p

text: for

text: all

text: n

text: is

text: suﬃciently

text: large.

text: Thus

text: we

text: only

text: need

text: to

text: establish

text: the

text: probability

text: bound

text: for

text: each

text: term

text: on

text: the

text: right

text: hand

text: side

text: of

text: First

text: consider

text: max

text: T

text: E(T

text: Using

text: similar

text: arguments

text: as

text: for

text: proving

text: j

text: p


text: we have | I | ≤ | |(cid:107) (cid:107) | |(cid:107) (cid:107) I ≤ X (xT Bβ B + zT Iγ I) C ( X x B + X z I ) C (s M2 + s M3). ij i, 0, i, 0, Φi 0 ij i, 1 ij i, 1 Φi 0 2 6 1 6 (cid:20) (cid:21) (cid:20) (cid:21) For any δ > 0, an application of Hoeﬀding’s inequality (Hoeﬀding, 1963) gives | − | ≥ ≤ (cid:18)− nδ(cid:19)2 (cid:18) ≤ −(cid:19) nδ2 P ( T E(T ) δ) 2 exp 2 exp j1 j1 2C2M4(s + s M )2 4C2M4(s2 + s2M2) 0 6 2 1 6 0 6 2 1 6 ≤ − nδ2 − nδ2 2 exp + 2 exp , 8C2M4s2 8C2M6s2 0 6 2 0 6 1 ≤ where we have used the fact that (a + b)2 2(a2 + b2) for any real numbers a and b, and − ≤ − − exp[ c/(a + b)] exp[ c/(2a)] + exp[ c/(2b)] for any a, b, c > 0. Recall that M6 = nη6. − Under Condition 1, taking δ = Cn κ2/6 results in (cid:88) p | − | ≥ − ≤ | − | ≥ − P ( max(cid:16) T E(T ) C(cid:17)n κ2/6) (cid:16) P ( T E(T (cid:17)) Cn κ2/6) ≤ ≤ (cid:101)j1 j1 (cid:101) j1 j1 1 j p j=1 ≤ − − − − − − − − 2p exp Cn1 2κ2 4η6 2ξ2 + 2p exp Cn1 2κ2 6η6 2ξ1 . (D.46) | | | | ≥ Next, consider max ≤ ≤ T . By Markov’s inequality, for any δ > 0, we have P ( T 1 j p j2 j2 ≤ − | | − δ) δ 1E( Tj2 ). In view of the ﬁrst inequality in (D.44), taking δ = Cn κ2/6 gives that (cid:101) | | ≥ − ≤ − − P ( T Cn κ2/6) 6C 1C Cnκ2(nξ1 + nξ2)(1 + nξ2 + 2nξ1)1/2 exp[ nα1η6/(2c )] j2 0 1 ≤ ≤ for all 1 j p. Therefore, (cid:88) p | | ≥ − ≤ | | ≥ − P ( max T Cn κ2/6) P ( T Cn κ2/6) ≤ ≤ (cid:101)j2 j2 1 j p j=1 ≤ − − 6pC 1C Cnκ2(nξ1 + nξ2)(1 + nξ2 + 2nξ1)1/2 exp[ nα1η6/(2c )]. (D.47) 0 1 Combining (D.45), (D.46), and (D.47) yields that for suﬃciently large n, 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: we

text: have

text: I

text: I

text: X

text: (xT

text: Bβ

text: B

text: zT

text: Iγ

text: I)

text: C

text: X

text: x

text: B

text: X

text: z

text: I

text: C

text: (s

text: s

text: ij

text: i,

text: i,

text: Φi

text: ij

text: i,

text: ij

text: i,

text: Φi

text: For

text: any

text: δ

text: an

text: application

text: of

text: Hoeﬀding’s

text: inequality

text: (Hoeﬀding,

text: gives

text: P

text: T

text: E(T

text: δ)

text: exp

text: exp

text: s

text: M

text: exp

text: exp

text: where

text: we

text: have

text: used

text: the

text: fact

text: that

text: (a

text: for

text: any

text: real

text: numbers

text: a

text: and

text: b,

text: and

text: exp[

text: c/(a

text: b)]

text: exp[

text: exp[

text: for

text: any

text: a,

text: b,

text: c

text: Recall

text: that

text: Under

text: Condition

text: taking

text: δ

text: Cn

text: results

text: in

text: p

text: P

text: T

text: E(T

text: P

text: T

text: E(T

text: Cn

text: j

text: p

text: exp

text: exp

text: Next,

text: consider

text: max

text: T

text: By

text: Markov’s

text: inequality,

text: for

text: any

text: δ

text: we

text: have

text: P

text: T

text: j

text: p

text: δ)

text: δ

text: In

text: view

text: of

text: the

text: ﬁrst

text: inequality

text: in

text: taking

text: δ

text: Cn

text: gives

text: that

text: P

text: T

text: Cn

text: exp[

text: for

text: all

text: j

text: p.

text: Therefore,

text: p

text: P

text: max

text: T

text: Cn

text: P

text: T

text: Cn

text: j

text: p

text: exp[

text: Combining

text: and

text: yields

text: that

text: for

text: suﬃciently

text: large

text: n,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:101) (cid:101) Following similar arguments as for proving (D.10), it can be shown that there exist some constants C > 0 and C > 0 such that for all n suﬃciently large, 1 2 (cid:101) (cid:101) | − | ≥ − − ≤ − { } P (max ωˆ ω 2 1c n κ1) 2s C exp[ C nmin α1,α2 r1]. (D.62) ∈A k k 2 1 1 2 k − Note that the right hand side of (D.62) can be bounded by o(n C1) for some arbitrarily large positive constant C . This gives 1 (cid:98) A ⊂ A ≥ − − P ( ) 1 o(n C1). (D.63) Thus combining (D.61) and (D.63) yields (cid:98) I ⊂ I ≥ − − P ( ) 1 o(n C1). (D.64) (cid:101) (cid:101) Using similar arguments as for proving part b) of Theorem 1 and (D.63), we can show that there exist some positive constants C , C , and C such that for all n suﬃciently large, 1 2 2 (cid:98) (cid:101) (cid:101) B ⊂ B ≥ | ∗ − ∗| − − ≥ − − P ( ) P (max ωˆ ω < 2 1c n κ2) 1 s C exp( C nα1r2) ∈B j j 2 2 1 2 j ≥ − − 1 o(n C2), (D.65) Combining (D.63) and (D.65) leads to (cid:99) (cid:98) (cid:98) (cid:98) (cid:98) M ⊂ M ≥ A ⊂ A B ⊂ B ≥ A ⊂ A B ⊂ B − P ( ) P ( and ) P ( ) + P ( ) 1 ≥ − − { } 1 o(n min C1,C2 ). (D.66) 33

text: Following

text: similar

text: arguments

text: as

text: for

text: proving

text: it

text: can

text: be

text: shown

text: that

text: there

text: exist

text: some

text: constants

text: C

text: and

text: C

text: such

text: that

text: for

text: all

text: n

text: suﬃciently

text: large,

label: equation

text: P

text: (max

text: ωˆ

text: ω

text: n

text: C

text: exp[

text: C

text: nmin

text: ∈A

text: k

text: k

text: k

text: Note

text: that

text: the

text: right

text: hand

text: side

text: of

text: can

text: be

text: bounded

text: by

text: o(n

text: for

text: some

text: arbitrarily

text: large

text: positive

text: constant

text: C

text: This

text: gives

text: A

text: A

text: P

text: o(n

text: Thus

text: combining

text: and

text: yields

text: I

text: I

text: P

text: o(n

text: Using

text: similar

text: arguments

text: as

text: for

text: proving

text: part

text: b)

text: of

text: Theorem

text: and

text: we

text: can

text: show

text: that

text: there

text: exist

text: some

text: positive

text: constants

text: C

text: C

text: and

text: C

text: such

text: that

text: for

text: all

text: n

text: suﬃciently

text: large,

text: B

text: B

text: P

text: P

text: (max

text: ωˆ

text: ω

text: n

text: s

text: C

text: exp(

text: C

text: ∈B

text: j

text: j

text: j

text: o(n

text: Combining

text: and

text: leads

text: to

text: M

text: M

text: A

text: A

text: B

text: B

text: A

text: A

text: B

text: B

text: P

text: P

text: and

text: P

text: P

text: o(n

text: min


text: (cid:101) (cid:101) ∈ D ⊂ { (cid:101)· · ·(cid:101) } n for each j 1, , p . Assume that for any constant C > 0, there exist constants · · · ≥ (cid:18) (cid:19) C , , C > 0 except C , C 0 such that (cid:110) (cid:111) (cid:110) (cid:111) 1 8 3 7 (cid:98) (cid:101) (cid:101) (cid:101) (cid:101) (cid:18) | − | ≥ − (cid:19) ≤ |D| (cid:110)− (cid:111) (cid:110)− (cid:111) P m∈aDx A(cid:98)j Aj Cn κ1 C(cid:101)1 exp C(cid:101)2nf(κ1) + C(cid:101)3 exp C(cid:101)4ng(κ1) , j | − | ≥ − ≤ |D| − − P max B B Cn κ1 C exp C nf(κ1) + C exp C ng(κ1) ∈D j j 5 6 7 8 j (cid:101) (cid:101) (cid:101) with f(κ ) and g(κ ) some functions of κ . Then for any constant C > 0, there exist 1 1 1 · · · ≥ (cid:26) (cid:27) constants C , , C > 0 except C 0 such that (cid:110) (cid:111) 9 12 11 (cid:98) (cid:98) (cid:101) (cid:101) | −(cid:110) − −(cid:111) | ≥ − ≤ |D| − P max(cid:101) (A B )(cid:101) (A B ) Cn κ1 C exp C nf(κ1) ∈D j j j j 9 10 j − + C exp C ng(κ1) . 11 12 (cid:98) (cid:98) (cid:98) (cid:98) | − − − | ≤ | − | | − | Proof. Note that max ∈D (A B ) (A B ) max ∈D A A +max ∈D B B . j j j j j j j j j j j Thus, for any positive constant C, (cid:98) (cid:98) | − − − | ≥ − P (max ((cid:98)A B ) (A B ) Cn κ1(cid:98)) ∈D j j j j j ≤ | (cid:110)− | ≥ (cid:111)− (cid:110) | −(cid:111) | ≥ − (cid:110) (cid:111) P (m(cid:101)ax A A(cid:101) Cn κ1/(cid:101)2) + P (m(cid:101)ax B B C(cid:101)n κ1/2) (cid:101) ∈D j j ∈D j j j j (cid:110) (cid:111) ≤|D| − − |D| − C(cid:101) exp C(cid:101) nf(κ1) + C exp C ng(κ1) + C exp C nf(κ1) 1 2 3 4 5 6 (cid:110) (cid:111) (cid:110) (cid:111) − + C(cid:101) exp C(cid:101) ng(κ1) (cid:101) (cid:101) 7 8 ≤|D| − − C exp C nf(κ1) + C exp C ng(κ1) , 9 10 11 12 (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) { } ≥ where C = C + C > 0, C = min C , C > 0, C = C + C 0, and C = 9 1 5 10 2 6 11 3 7 12 { } min C , C > 0. 4 8 (cid:98) ∈ D ⊂ { · · · } ≥ Lemma 10. Let B ’s with j 1, , p satisfy min ∈D B L for some constant j j j 4 (cid:101) (cid:101) ∈ D L > 0, and B be an estimate of B based on a sample of size n for each j . Assume 4 j j that for any constant C > 0, there exist constants C , C > 0 such that 1 2 (cid:18) (cid:19) (cid:110) (cid:111) (cid:98) (cid:101) (cid:101) | − | ≥ − ≤ |D| − P max B B Cn κ1 C exp C nf(κ1) . ∈D j j 1 2 j 50

text: D

text: n

text: for

text: each

text: j

text: p

text: Assume

text: that

text: for

text: any

text: constant

text: C

text: there

text: exist

text: constants

text: C

text: C

text: except

text: C

text: C

text: such

text: that

text: |D|

text: P

text: m∈aDx

text: Aj

text: Cn

text: κ1

text: exp

text: exp

text: j

text: |D|

text: P

text: max

text: B

text: B

text: Cn

text: κ1

text: C

text: exp

text: C

text: C

text: exp

text: C

text: ∈D

text: j

text: j

text: j

text: with

text: f(κ

text: and

text: g(κ

text: some

text: functions

text: of

text: κ

text: Then

text: for

text: any

text: constant

text: C

text: there

text: exist

text: constants

text: C

text: C

text: except

text: C

text: such

text: that

text: |D|

text: P

text: (A

text: B

text: (A

text: B

text: Cn

text: κ1

text: C

text: exp

text: C

text: ∈D

text: j

text: j

text: j

text: j

text: j

text: C

text: exp

text: C

text: Proof.

text: Note

text: that

text: max

text: ∈D

text: (A

text: B

text: (A

text: B

text: max

text: ∈D

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text: A

text: +max

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text: B

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text: Cn

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text: P

text: A

text: Cn

text: P

text: B

text: B

text: ∈D

text: j

text: j

text: ∈D

text: j

text: j

text: j

text: j

text: ≤|D|

text: |D|

text: exp

text: C

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text: exp

text: C

text: exp

text: ≤|D|

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text: C

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text: C

text: min

text: C

text: C

text: C

text: C

text: C

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text: C

text: min

text: C

text: C

text: D

text: Lemma

text: Let

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text: D

text: L

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text: (cid:98) (cid:113) (cid:98) (cid:112) | (cid:98) − | B B ≥ − | | − P (m∈aDx | j j | (cid:112)Cn κ1, m∈iDn(cid:112)Bj > L4 L0) (cid:110) (cid:111) j (cid:98) j (cid:101) (cid:101) B + B j j ≤ { | − | ≥ − − } ≤ |D| − P max B B C( L L + L )n κ1 C exp C nf(κ1) , (E.15) ∈D j j 4 0 4 5 6 j (cid:101) (cid:101) where C and C are some positive constants. Combining (E.13), (E.14), and (E.15) 5 6 gives (cid:113) (cid:112) (cid:110) (cid:111) (cid:98) (cid:101) (cid:101) | − | ≥ − ≤ |D| − P (max B B Cn κ1) C exp C nf(κ1) , (E.16) ∈D j j 3 4 j (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) { } where C = C + C and C = min C , C . 3 1 5 4 2 6 ∈ D ⊂ {(cid:98) · · · (cid:98)} | | ≤ | | ≤ Lemma 11. Let A ’s with j 1, , p and B satisfy max ∈D A L and B L j j j 5 6 for some constants L , L > 0, and A and B be estimates of A and B, respectively, based 5 6 j j (cid:101) (cid:101) (cid:101)∈ D on a sample of size n for each j . Assume that for any constant C > 0, there exist · · · ≥ (cid:18) (cid:19) constants C , , C > 0 except C 0 such tha(cid:110)t (cid:111) (cid:110) (cid:111) 1 8 3 (cid:98) (cid:101) (cid:101) (cid:101) (cid:101) (cid:16) | − | ≥ (cid:17) − ≤ |(cid:110) | −(cid:111) (cid:110) (cid:111)− P m(cid:98)ax A A Cn κ(cid:101)1 D C(cid:101) exp C n(cid:101)f(κ1) + C(cid:101) exp C ng(κ1) , ∈D j j 1 2 3 4 j | − | ≥ − ≤ − − P B B Cn κ1 C exp C nf(κ1) + C exp C ng(κ1) 5 6 7 8 (cid:101) (cid:101) with f(κ ) and g(κ ) some functions of κ . Then for any constant C > 0, there exist 1 1 1 · · · constants C , , C > 0 such that 9 12 (cid:18) (cid:19) (cid:110) (cid:111) (cid:110) (cid:111) (cid:98) (cid:98) (cid:101) (cid:101) (cid:101) (cid:101) | − | ≥ − ≤ | | − − P max A B A B Cn κ1 D C exp C nf(κ1) + C exp C ng(κ1) . ∈D j j 9 10 11 12 j (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) | − | ≤ | − | | − | Proof. Note that max ∈D A B A B max ∈D A (B B) + max ∈D (A A )B . j j j j j j j j 52 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: B

text: B

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text: j

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text: B

text: B

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text: j

text: |D|

text: P

text: max

text: B

text: B

text: C(

text: L

text: L

text: L

text: )n

text: κ1

text: C

text: exp

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text: j

text: j

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text: C

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text: C

text: are

text: some

text: positive

text: constants.

text: Combining

text: and

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text: C

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text: C

text: and

text: C

text: min

text: C

text: C

text: D

text: Lemma

text: Let

text: A

text: ’s

text: with

text: j

text: p

text: and

text: B

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text: L

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text: L

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text: P

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text: A

text: Cn

text: D

text: exp

text: C

text: exp

text: C

text: ∈D

text: j

text: j

text: j

text: P

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text: B

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text: κ1

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text: C

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text: f(κ

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text: for

text: any

text: constant

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text: C

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text: P

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text: B

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text: C

text: exp

text: C

text: C

text: exp

text: C

text: ∈D

text: j

text: j

text: j

text: Proof.

text: Note

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text: B

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