text: FLEXIBLE BEHAVIORAL CAPTURE-RECAPTURE MODELLING 13 Table 3. Great Copper Butterﬂy data: point and interval estimates together − with AIC index of alternative ﬁtted models. Conﬁdence intervals at level 1 α = 0.95. We note that model M corresponding to the optimal single cut ∗ z.cut(1) e = 0.625 corresponds to model M described in Farcomeni (2011) 1 L2 − Model # parameters Nˆ (N , N+) AIC M 1+2 170 (87,448) 321.46 z M 1+5 62 (48,223) 321.54 z.cut(4) M 1+4 62 (48,223) 321.62 z.cut(3) M 1+3 176 (78,243) 323.36 z.cut(2) M 1+5 62 (48,223) 325.46 c2b M 1+2 154 (82,367) 325.99 zgn M 1+2 90 (63,152) 326.01 z.cut(1) M 1+4 176 (78,896) 327.20 c2 M 1+2 97 (64,181) 330.93 c1 M 1+3 62 (48,223) 331.24 c1b M 1+2 96 (62,184) 338.30 zg˜n M 1+1 64 (53,85) 342.80 0 M 1+2 68 (54,97) 343.77 zf M 1+2 62 (48,223) 344.77 b M 1+8 64 (52,84) 352.85 t there may remain some doubts on the ability of detecting the right longitudinal pattern with so few captured individuals during a moderate number of trapping occasions. This issue will be addressed in Section . Giant Day Geckos. The giant day gecko (Phelsuma madagascariensis grandis) is a tropical reptile living in areas of tropical and subtropical forest in northern Madagascar. A capture- recapture sampling on the giant day gecko has been conducted in the Masoala rainforest exhibit at the Zurich Zoo and the resulting data have been analyzed in Wanger et al. (2009). Due to the high number of capture occasions (t = 30) it can be considered an unusually good dataset where the closed population assumption is valid since it is a captive population. We are interested in analyzing behavioural patterns possibly originated by feed habits of the geckos and/or by the human presence. More details on the sampling process can be found in Wanger et al. (2009). In Table 4 we list the results obtained by ﬁtting a collection of standard models as well as new models based on both the originally proposed g(x) and g (x). With this data set we found n that the alternative behavioural covariate z = g (x) based on the number of previous captures n as deﬁned in (9) allows to achieve a better ﬁt. In this example the AIC index highlights a diﬀerent kind of behavioural response selecting as best model M which partitions zgn.cut(3) with three cutpoints the behavioural covariate range in four subintervals [0, 0.05], (0.05, 0.1579], (0.1579, 0.625] and (0.625, 1] with corresponding conditional probability estimates pˆ = 0.034, H1 pˆ = 0.061, pˆ = 0.159 and pˆ = 0.375. The optimal cuts have been determined by a grid H2 H3 H4 search as detailed in the Supplementary Web Materials (S3). 6. A simulation study for model selection In this section a simulation study is developed in order to evaluate and compare the per- formance among alternative classical (M , M , M , M , M , M and M ) and new (M , 0 b t c1 c1b c2 c2b z M , M and M ) models based on four meaningful behavioural covariates. zgn zf zg˜n ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: The

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text: FLEXIBLE BEHAVIORAL CAPTURE-RECAPTURE MODELLING 16 c c Trial3 rmseCIl%aiCI7.69529.295 13.93416.40 21.704.60 17.8711.40 22.103.90 15.39079.30 12.45018.40 15.39079.30 9.18425.34 15.39079.31 21.703.90 Trial6 rmseCIl%aiCI11.39442.197 25.21423.90 41.608.20 33.5116.90 42.507.00 18.49486.80 21.63126.80 18.49486.80 13.58236.22 18.49486.81 42.506.90 )wheretruegenerating denceintervalscoverage modelhasachievedbest Tardella,2013).Weare−ceinterval1α=0.95. al2¯l%aicNCI52.68899 28.8087 8.7079 19.1083 7.1078 1776.5*098 30.2089 1776.5*098 47.0994 1776.5*398 6.9078 al5¯l%aicNCI72.996201 39.90176 12.80159 26.00167 10.90158 942.1*0197 41.90180 942.1*0197 62.61191 942.1*3197 10.70158 imulationsetting(Trial1-6 ansquareerror(rmse),conﬁ ageoftimeseachcompeting∞=(AlunniFegatelliand Nominalleveloftheconﬁden Tri rmseCI 13.995 20.847 33.40 27.310 34.30 105.4*89 19.655 105.4*89 15.986 105.4*89 34.30 Tri rmseCI 19.891 39.519 67.10 54.50 68.90 130.7*93 37.326 130.7*93 24.777 130.7*93 68.90 etsforeachsˆteN,rootme )andpercentˆoodfailureN ailurecases. ¯aicN 75100 282 067 074 066 0119* 383 0119* 1493 6119* 066 ¯aicN 87199 0163 0133 0146 0131 0218* 0165 0218* 11183 2218* 0131 mulateddatas epointestima intervals(lCI enceoflikelih movingthosef Trial1 CIl%CI90187.1 6678.8 117.1 2140.3 011.1 833866.0* 6282.1 833866.0* 89151.7* 833866.0* 010.7 Trial4 CIl%CI95200.0 4591.5 021.6 748.5 014.9 873270.3* 3485.6 873270.3* 85192.2 873270.3* 014.6 studywith100si¯average(N)ofth oftheconﬁdence ndenotesthepres computedafterre rmse 43.0 33.9 53.3 45.2 56.1 564.0* 34.7 564.0* 46.2* 564.0* 56.4 rmse 45.9 62.8 107.4 90.3 112.8 415.9* 68.4 415.9* 53.1 415.9* 113.0 mulation empirical agelength∗Thesig nitermse ¯N 109 75 47 57 44 209* 72 209* 96* 209* 44 ¯N 205 142 93 112 88 284* 136 284* 174 284* 876.Si M:zaver aic). gaﬁ s , % n el el le eli %) ( rti Mod MzMzgnMzfMz˜gnM0MbMc1Mcb1Mc2Mcb2Mt Mod MzMzgnMzfMz˜gnM0MbMc1Mcb1Mc2Mcb2MtTab mod (CI AIC repo ##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: FLEXIBLE BEHAVIORAL CAPTURE-RECAPTURE MODELLING 18 Unconditional likelihood inference is easily implemented recycling consolidated standard GLM routines. An integrated suite of R (R Core Team, 2013) functions have been developed and are available as an R package upon request. We hint also at a possible extension outside the closed capture-recapture context of the quantization idea. The same idea can be applied more generally to model memory eﬀects in studies with longitudinal binary outcomes where binary events such as successful surgery experiences or correctly performed tasks are observed. Also it is possible to generalize this strategy to categorical-ordinal data using an appropriate scaling. There are certainly other issues which should be addressed for a more thorough understanding of real data such as allowing for heterogeneous capture probabilities and including the possible presence of individual covariates. While the former aspect can be easily accommodated and implemented as already argued within the standard GLM framework through the addition of a longitudinal individual random eﬀect the latter is more diﬃcult to be embedded in the proposed inferential setting which uses the unconditional likelihood. In this case individual covariates would not be available for unobserved units. Indeed possible alternative ways out are the use of conditional likelihood or implementing our models using data augmentation within a Bayesian framework following the approach in Royle (2009). We actually plan to develop this in a future work. Actually some previous work on alternative inferential approaches for standard behavioural models Alunni Fegatelli and Tardella (2013) suggests that the Bayesian approach could be more promising. Here we have focussed more speciﬁcally on understanding the role, meaning and possible alternative uses of the new memory-eﬀect covariates and their connections with already available models. References J. M. Alho. Logistic regression in capture-recapture models. Biometrics, 46:623–635, 1990. D. Alunni Fegatelli and L. Tardella. Improved inference on capture recapture models with behavioural eﬀects. Statistical Methods & Applications, 22(1):45–66, 2013. S. C. Amstrup, T. L. McDonald, and B. F. Manly, editors. Handbook of Capture-Recapture Analysis. Princeton University Press, 2005. F. Bartolucci and F. Pennoni. A Class of Latent Markov Models for Capture-Recapture Data Allowing for Time, Heterogeneity, and Behavior Eﬀects. Biometrics, 63(2):568–578, 2007. D. B¨ohning. Editorial - recent developments in capture-recapture methods and their applica- tions. Biometrical Journal, 50(6):954–956, 2008. D. Borchers, S. Buckland, and W. Zucchini. Estimating Animal Abundance: Closed Populations. Statistics for Biology and Health. Springer, 2002. A. Chao, W. Chu, and C.-H. Hsu. Capture-recapture when time and behavioral response aﬀect capture probabilities. Biometrics, 56(2):427–433, 2000. A. Farcomeni and D. Scacciatelli. Heterogeneity and behavioral response in continuous time capture-recapture, with application to street cannabis use in italy. to appear on the Annals of Applied Statistics, 2014. A. Farcomeni. Recapture models under equality constraints for the conditional capture proba- bilities. Biometrika, 2011. S. K. Ghosh and J. L. Norris. Bayesian capture-recapture analysis and model selection allowing for heterogeneity and behavioral eﬀects. Journal of Agricultural, Biological, and Environmen- tal Statistics, 10(1):35–49, 2005. R. M. Huggins. On the statistical analysis of capture experiments. Biometrika, 76:133–140, 1989.

text: FLEXIBLE

text: BEHAVIORAL

text: CAPTURE-RECAPTURE

text: MODELLING

text: Unconditional

text: likelihood

text: inference

text: is

text: easily

text: implemented

text: recycling

text: consolidated

text: standard

text: GLM

text: routines.

text: An

text: integrated

text: suite

text: of

text: R

text: (R

text: Core

text: Team,

text: functions

text: have

text: been

text: developed

text: and

text: are

text: available

text: as

text: an

text: R

text: package

text: upon

text: request.

text: We

text: hint

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text: at

text: a

text: possible

text: extension

text: outside

text: the

text: closed

text: capture-recapture

text: context

text: of

text: the

text: quantization

text: idea.

text: The

text: same

text: idea

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text: be

text: applied

text: more

text: generally

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text: model

text: memory

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text: in

text: studies

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text: capture

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text: individual

text: covariates.

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text: the

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text: unconditional

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text: In

text: this

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text: would

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text: Indeed

text: possible

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text: the

text: use

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text: or

text: implementing

text: our

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text: Bayesian

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text: following

text: the

text: approach

text: in

text: Royle

text: We

text: actually

text: plan

text: to

text: develop

text: this

text: in

text: a

text: future

text: work.

text: Actually

text: some

text: previous

text: work

text: on

text: alternative

text: inferential

text: approaches

text: for

text: standard

text: behavioural

text: models

text: Alunni

text: Fegatelli

text: and

text: Tardella

text: suggests

text: that

text: the

text: Bayesian

text: approach

text: could

text: be

text: more

text: promising.

text: Here

text: we

text: have

text: focussed

text: more

text: speciﬁcally

text: on

text: understanding

text: the

text: role,

text: meaning

text: and

text: possible

text: alternative

text: uses

text: of

text: the

text: new

text: memory-eﬀect

text: covariates

text: and

text: their

text: connections

text: with

text: already

text: available

text: models.

text: References

text: J.

text: M.

text: Alho.

text: Logistic

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text: models.

text: Biometrics,

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text: L.

text: Tardella.

text: Improved

text: inference

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text: BEHAVIORAL

text: CAPTURE-RECAPTURE

text: MODELLING

text: W.-H.

label: reference

text: Hwang

text: and

text: R.

text: Huggins.

text: A

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text: functional

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text: capture-recapture

text: experiments.

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text: New

text: Zealand

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text: W.-H.

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text: P.

text: S.

text: F.

text: Yip.

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text: (gekkonidae,

text: phelsuma

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text: grandis)

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text: masoala

text: rainforest

text: exhibit,

text: zurich

text: zoo.

text: Ecological

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text: C.

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text: D.

text: R.

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text: K.

text: P.

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text: FLEXIBLE BEHAVIORAL CAPTURE-RECAPTURE MODELLING 25 − augmented history x = (0, .., 0, x) with k l zeroes ahead as follows aug x g (x) = g(x ). aug aug This can be formalized in matrix notation considering the whole binary capture history matrix X and deriving an augmented matrix X = [0, ..., 0, X] obtained by adding k columns of zeros aug on the left side of the original matrix X. One can compute in the usual way the corresponding covariate matrix Z by applying the original function g to all partial capture histories in X . aug aug At this point, instead of the former matrix Z built directly from X one uses as covariate matrix only the last t columns of Z . aug S3. Models driven by partitions of the range of meaningful behavioural covariates and the search of optimal partitions Let us now show that model M proposed in Farcomeni (2011) can be recovered within our L2 general logistic regression framework which relies on the meaningful numeric covariate z adopting as a regression function a step function with only two levels corresponding to the bipartition of the range of z into two contiguous intervals: [0, 0.625]; (0.625, 1]. In fact, for the initial ≥ partial capture histories (1), (01) and for all other x with lx 3 such that (x1, . . . , xlx−3, 1, 0, 1), (x1, . . . , xlx−3, 0, 1, 1) and (x1, . . . , xlx−3, 1, 1, 1) we have that z > 0.625. Th(cid:0)is can b(cid:3)e easily checked numerically for the ﬁrst two histories x = (1) and x = (01) since g(x ) = 1 and 1 2 1 (cid:80) g(x ) = 2/3. For all the other partial capture histories we can focus on the last three digits. We 2 − have already argued that there are 8 = 23 subintervals I = [0, 1/23] and I = r 1, r for any r(cid:80)= 1, ..., 23 such that z ∈ I if and only if 3 x − 21p−1 = r−1. Since (r0.625,213] =23I ∪I ∪I r p=1 lx 3+p 6 7 8 one can easily v−erify nu−mer≥ically that the only three last digits (xlx−2, xlx−1, xlx) such that 3 x − 2p 1 = r 1 5 are (1, 0, 1), (0, 1, 1) and (1, 1, 1). Hence the intervals [0, 0.625] p=1 lx 3+p (cid:0) (cid:3) and (0.625, 1] lead to the same bipartition of the set H considered in model M . We also remark L2 that in light of our argument there is an underlying correspondence between model M and − L2 Markovian models of order k = 3 related to partition intervals I = r 1, r of the behavioural r 23 23 covariate z = g(x). In fact we can regard model M as a simpliﬁed reduced Markovian model L2 of order k = 3. Indeed once acknowledged that model M corresponds to one of the possible bipartition L2 of the range of z one can wonder whether there are other bipartitions which can ﬁt the data better. This naturally leads us to look for an optimal bipartition of the range in terms of the AIC resulting from the corresponding model. We considered models associated to alternative ∪ ∗ ∈ intervals [0, e ] (e , 1] and eventually determine the best cutpoint e [0, 1] denoting the 1 1 H 1 { } corresponding bipartition of partial capture histories with (M ) = H , H where H 2 z.cut(1) 1 2 1 and H are such that 2 ∈ ⇐⇒ ∈ ∗ ∈ ⇐⇒ ∈ ∗ x H g(x) [0, e ] ; x H g(x) (e , 1] 1 1 2 1 In fact, in our applications we found the optimal cutpoint through a simple ﬁnite grid search among all the values e = g(x) corresponding to an actually observed partial capture history 1 x. We remark that the optimal single cut found in the Great Copper example ends up being ∗ e = 0.625 which actually corresponds to model M . 1 L2 Moreover, this idea can be extended to more than one cutpoint. In this case the computa- tional burden for the ﬁnite grid search with cutpoints corresponding to actually observed partial capture histories becomes heavier. Despite that in all our real data applications and simulations we were able to easily implement the full search up to two cutpoints. For more than two cut- points we considered two alternative strategies: performing a simpliﬁed search reducing the set of possible cuts to a subset of quantiﬁcations of actually observed partial capture histories or ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FLEXIBLE

text: BEHAVIORAL

text: CAPTURE-RECAPTURE

text: MODELLING

text: augmented

text: history

text: x

text: x)

text: with

text: k

text: l

text: zeroes

text: ahead

text: as

text: follows

text: aug

text: x

text: g

text: (x)

text: g(x

text: aug

text: aug

text: This

text: can

text: be

text: formalized

text: in

text: matrix

text: notation

text: considering

text: the

text: whole

text: binary

text: capture

text: history

text: matrix

text: X

text: and

text: deriving

text: an

text: augmented

text: matrix

text: X

text: X]

text: obtained

text: by

text: adding

text: k

text: columns

text: of

text: zeros

text: aug

text: on

text: the

text: left

text: side

text: of

text: the

text: original

text: matrix

text: X.

text: One

text: can

text: compute

text: in

text: the

text: usual

text: way

text: the

text: corresponding

text: covariate

text: matrix

text: Z

text: by

text: applying

text: the

text: original

text: function

text: g

text: to

text: all

text: partial

text: capture

text: histories

text: in

text: X

text: aug

text: aug

text: At

text: this

text: point,

text: instead

text: of

text: the

text: former

text: matrix

text: Z

text: built

text: directly

text: from

text: X

text: one

text: uses

text: as

text: covariate

text: matrix

text: only

text: the

text: last

text: t

text: columns

text: of

text: Z

text: aug

text: Models

text: driven

text: by

text: partitions

text: of

text: the

text: range

text: of

text: meaningful

text: behavioural

text: covariates

text: and

text: the

text: search

text: of

text: optimal

text: partitions

text: Let

text: us

text: now

text: show

text: that

text: model

text: M

text: proposed

text: in

text: Farcomeni

text: can

text: be

text: recovered

text: within

text: our

text: general

text: logistic

text: regression

text: framework

text: which

text: relies

text: on

text: the

text: meaningful

text: numeric

text: covariate

text: z

text: adopting

text: as

text: a

text: regression

text: function

text: a

text: step

text: function

text: with

text: only

text: two

text: levels

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text: to

text: the

text: bipartition

text: of

text: the

text: range

text: of

text: z

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text: two

text: contiguous

text: intervals:

text: In

text: fact,

text: for

text: the

text: initial

text: partial

text: capture

text: histories

text: and

text: for

text: all

text: other

text: x

text: with

text: lx

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text: that

text: and

text: we

text: have

text: that

text: z

text: can

text: easily

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text: numerically

text: for

text: the

text: ﬁrst

text: two

text: histories

text: x

text: and

text: x

text: since

text: g(x

text: and

text: g(x

text: For

text: all

text: the

text: other

text: partial

text: capture

text: histories

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text: can

text: focus

text: on

text: the

text: last

text: three

text: digits.

text: We

text: have

text: already

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text: that

text: there

text: are

text: subintervals

text: I

text: and

text: I

text: r

text: r

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text: z

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text: ∪I

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text: nu−mer≥ically

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text: only

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text: xlx)

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text: the

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text: lx

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text: lead

text: to

text: the

text: same

text: bipartition

text: of

text: the

text: set

text: H

text: considered

text: in

text: model

text: M

text: We

text: also

text: remark

text: that

text: in

text: light

text: of

text: our

text: argument

text: there

text: is

text: an

text: underlying

text: correspondence

text: between

text: model

text: M

text: and

text: Markovian

text: models

text: of

text: order

text: k

text: related

text: to

text: partition

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text: I

text: r

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text: the

text: behavioural

text: r

text: covariate

text: z

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text: In

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text: model

text: M

text: as

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text: z

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text: bipartition

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text: AIC

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text: the

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text: model.

text: We

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text: associated

text: to

text: alternative

text: intervals

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text: determine

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text: (M

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text: the

text: optimal

text: cutpoint

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text: a

text: simple

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text: search

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text: all

text: the

text: values

text: e

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text: observed

text: partial

text: capture

text: history

text: x.

text: We

text: remark

text: that

text: the

text: optimal

text: single

text: cut

text: found

text: in

text: the

text: Great

text: Copper

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text: up

text: being

text: e

text: which

text: actually

text: corresponds

text: to

text: model

text: M

text: Moreover,

text: this

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text: be

text: extended

text: to

text: more

text: than

text: one

text: cutpoint.

text: In

text: this

text: case

text: the

text: computa-

text: tional

text: burden

text: for

text: the

text: ﬁnite

text: grid

text: search

text: with

text: cutpoints

text: corresponding

text: to

text: actually

text: observed

text: partial

text: capture

text: histories

text: becomes

text: heavier.

text: Despite

text: that

text: in

text: all

text: our

text: real

text: data

text: applications

text: and

text: simulations

text: we

text: were

text: able

text: to

text: easily

text: implement

text: the

text: full

text: search

text: up

text: to

text: two

text: cutpoints.

text: For

text: more

text: than

text: two

text: cut-

text: points

text: we

text: considered

text: two

text: alternative

text: strategies:

text: performing

text: a

text: simpliﬁed

text: search

text: reducing

text: the

text: set

text: of

text: possible

text: cuts

text: to

text: a

text: subset

text: of

text: quantiﬁcations

text: of

text: actually

text: observed

text: partial

text: capture

text: histories

text: or

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FLEXIBLE BEHAVIORAL CAPTURE-RECAPTURE MODELLING 4 likelihood provided that Pˆ is jointly determined with Nˆ according to the deﬁnition of the 0 unconditional (or complete) likelihood. From the saturated parametrization based on p one can specify a parsimonious nested model based on a suitable partition of the conditional probabilities in p in terms of equivalence classes. { Let H be the set of all partial capture histories: H = () , (0), (1), (00), (10), (01), (11), } ∪ − X X { } H . . . = t 1 j where 0 = () . Let us denote with one of the possible partitions of H in j=0 B B disjoint subsets H { } = H , . . . , H , . . . , H B 1 b B ⊂ where each H H. The role of the index set H is to list all the partial capture histories b which may yield possible changes in the conditional capture probability depending on the past. Let us denote a generic partial capture history as follows x = (x , . . . , x ) where l is the length H 1 lx x of the binary vector. For each partition we consider a corresponding reduced parameter B vector of probabilities denoted with pH = (p , . . . , p ). The partition of capture histories B H1 HB in equivalence classes is such that ∀ (cid:48) ∈ ⇒ (cid:48) ∀ x, x Hb plx+1(x) = plx(cid:48)+1(x ) = pHb b = 1, . . . , B Notice that when there is absence of previous capture history (x = ()) we have l = 0. H x With the partition of subsets of H representing equivalence classes we have just reviewed B under a more convenient formalization the linear constrained approach in Farcomeni (2011) which indeed can be seen as stemmed from the works of Huggins (1989) and Alho (1990). As a simple example of our formalization based on partitions of subsets of H as opposed to the linear constrained approach one can consider model M . Indeed it can be deﬁned using two b (cid:40) blocks of equality constraints · · · p () = p (0) = p (0, 0) = = p (0, . . . , 0) = π M : 1 2 3 · · · t f b p (1) = p (1, 0) = p (0, 1) = = p (1, . . . , 1) = π 2 3 3 t r It is easy to verify that if we interpret π as the probability of ﬁrst capture and π as f r the probability of being recaptured one gets the most simple form of behavioural model with enduring eﬀect usually denoted with M . Equivalently, in our partition notation, model M corresponds to(cid:40)a bipartition H (M ) = {bH , H } such th(cid:110)at (cid:80) (cid:111) b 2 b 1 2 { } X ∪ ∈ ∪ − X H H = (), (0), (00), . . . , (0 . . . 0) = 0 x t 1 j : lx x = 0 2(Mb) = 1 \ j=1 j=1 j (2) H = H H 2 1 and the vector of parameters (π , π ) is represented in our notation as pH = (p , p ). f r 2(Mb) H1 H2 In Farcomeni (2011) is also shown that many other models proposed in the literature such as model M , M , M , M can be recovered as special cases of model with saturated pa- 0 ck ckb t rameterization p subject to speciﬁc linear constraints. In the following we prefer to index parameters with the partition notation and we refer to the reduced parametrization with the symbol pH = (p , . . . , p ) corresponding to the uniquely identiﬁed conditional probabilities B H1 HHB associated to the partition . B We now brieﬂy provide details on two other meaningful partitions corresponding to Markovian models of order 1 and 2 respectively. They will be used and clariﬁed more extensively in the next

text: FLEXIBLE

text: BEHAVIORAL

text: CAPTURE-RECAPTURE

text: MODELLING

text: likelihood

text: provided

text: that

text: Pˆ

text: is

text: jointly

text: determined

text: with

text: Nˆ

text: according

text: to

text: the

text: deﬁnition

text: of

text: the

text: unconditional

text: (or

text: complete)

text: likelihood.

text: From

text: the

text: saturated

text: parametrization

text: based

text: on

text: p

text: one

text: can

text: specify

text: a

text: parsimonious

text: nested

text: model

text: based

text: on

text: a

text: suitable

text: partition

text: of

text: the

text: conditional

text: probabilities

text: in

text: p

text: in

text: terms

text: of

text: equivalence

text: classes.

text: Let

text: H

text: be

text: the

text: set

text: of

text: all

text: partial

text: capture

text: histories:

text: H

text: X

text: X

text: H

text: t

text: j

text: where

text: Let

text: us

text: denote

text: with

text: one

text: of

text: the

text: possible

text: partitions

text: of

text: H

text: in

text: B

text: B

text: disjoint

text: subsets

text: H

text: H

text: H

text: H

text: B

text: b

text: B

text: where

text: each

text: H

text: H.

text: The

text: role

text: of

text: the

text: index

text: set

text: H

text: is

text: to

text: list

text: all

text: the

text: partial

text: capture

text: histories

text: b

text: which

text: may

text: yield

text: possible

text: changes

text: in

text: the

text: conditional

text: capture

text: probability

text: depending

text: on

text: the

text: past.

text: Let

text: us

text: denote

text: a

text: generic

text: partial

text: capture

text: history

text: as

text: follows

text: x

text: (x

text: x

text: where

text: l

text: is

text: the

text: length

text: H

text: lx

text: x

text: of

text: the

text: binary

text: vector.

text: For

text: each

text: partition

text: we

text: consider

text: a

text: corresponding

text: reduced

text: parameter

text: B

text: vector

text: of

text: probabilities

text: denoted

text: with

text: pH

text: (p

text: p

text: The

text: partition

text: of

text: capture

text: histories

text: B

text: HB

text: in

text: equivalence

text: classes

text: is

text: such

text: that

text: x,

text: x

text: Hb

text: pHb

text: b

text: B

text: Notice

text: that

text: when

text: there

text: is

text: absence

text: of

text: previous

text: capture

text: history

text: (x

text: we

text: have

text: l

text: H

text: x

text: With

text: the

text: partition

text: of

text: subsets

text: of

text: H

text: representing

text: equivalence

text: classes

text: we

text: have

text: just

text: reviewed

text: B

text: under

text: a

text: more

text: convenient

text: formalization

text: the

text: linear

text: constrained

text: approach

text: in

text: Farcomeni

text: which

text: indeed

text: can

text: be

text: seen

text: as

text: stemmed

text: from

text: the

text: works

text: of

text: Huggins

text: and

text: Alho

text: As

text: a

text: simple

text: example

text: of

text: our

text: formalization

text: based

text: on

text: partitions

text: of

text: subsets

text: of

text: H

text: as

text: opposed

text: to

text: the

text: linear

text: constrained

text: approach

text: one

text: can

text: consider

text: model

text: M

text: Indeed

text: it

text: can

text: be

text: deﬁned

text: using

text: two

text: b

text: blocks

text: of

text: equality

text: constraints

text: p

text: p

text: p

text: p

text: π

text: M

text: t

text: f

text: b

text: p

text: p

text: p

text: p

text: π

text: t

text: r

text: It

text: is

text: easy

text: to

text: verify

text: that

text: if

text: we

text: interpret

text: π

text: as

text: the

text: probability

text: of

text: ﬁrst

text: capture

text: and

text: π

text: as

text: f

text: r

text: the

text: probability

text: of

text: being

text: recaptured

text: one

text: gets

text: the

text: most

text: simple

text: form

text: of

text: behavioural

text: model

text: with

text: enduring

text: eﬀect

text: usually

text: denoted

text: with

text: M

text: Equivalently,

text: in

text: our

text: partition

text: notation,

text: model

text: M

text: corresponds

text: bipartition

text: H

text: (M

text: {bH

text: H

text: such

text: b

text: b

text: X

text: X

text: H

text: H

text: x

text: t

text: j

text: lx

text: x

text: j

text: H

text: H

text: H

text: and

text: the

text: vector

text: of

text: parameters

text: (π

text: π

text: is

text: represented

text: in

text: our

text: notation

text: as

text: pH

text: (p

text: p

text: f

text: r

text: In

text: Farcomeni

text: is

text: also

text: shown

text: that

text: many

text: other

text: models

text: proposed

text: in

text: the

text: literature

text: such

text: as

text: model

text: M

text: M

text: M

text: M

text: can

text: be

text: recovered

text: as

text: special

text: cases

text: of

text: model

text: with

text: saturated

text: pa-

text: ck

text: ckb

text: t

text: rameterization

text: p

text: subject

text: to

text: speciﬁc

text: linear

text: constraints.

text: In

text: the

text: following

text: we

text: prefer

text: to

text: index

text: parameters

text: with

text: the

text: partition

text: notation

text: and

text: we

text: refer

text: to

text: the

text: reduced

text: parametrization

text: with

text: the

text: symbol

text: pH

text: (p

text: p

text: corresponding

text: to

text: the

text: uniquely

text: identiﬁed

text: conditional

text: probabilities

text: B

text: HHB

text: associated

text: to

text: the

text: partition

text: B

text: We

text: now

text: brieﬂy

text: provide

text: details

text: on

text: two

text: other

text: meaningful

text: partitions

text: corresponding

text: to

text: Markovian

text: models

text: of

text: order

text: and

text: respectively.

text: They

text: will

text: be

text: used

text: and

text: clariﬁed

text: more

text: extensively

text: in

text: the

text: next


text: FLEXIBLE BEHAVIORAL CAPTURE-RECAPTURE MODELLING 10 { } that since 1/2t > 0 the ﬁrst partition I can be equivalently reduced to the single value 0 or 1 ≤ any other interval [0, e ] provided that e 1/2t. This is basically due to the discreteness of 1 1 the observable range. More generally, an alternative partition of the range of z into A consecutive subintervals I = [0, e ], . . . , I = (e − , e ], . . . , I = (e − , 1] represents a meaningful behavioural model 1 1 a a 1 a A A 1 corresponding to the regression step function ∀ ∈ s(z) = logit(p ) z I a = 1, . . . A. Ha a This particularly ﬂexible instance of partitioning the range of the behavioural covariate g(x) embeds some of the original models proposed in Farcomeni (2011) such as M (Supplementary L2 Web Materials, S1 and S3). In fact looking for an appropriate number and location of the partition cuts e in terms of step functions for the logistic regression can readily explore a a range of meaningful variable order Markov chain models. We will exploit this approach in our applications. As already mentioned the most critical parameter for the estimation of N is the probability P as in (1). Indeed when we partition the set of conditional probabilities through partitioning 0 ∈ the quantiﬁcation z = g(x) into intervals I , ... , I we have that, as long as g(x) I = [0, e ] 1 A 1 1 (and this is certainly true for all partial capture histories with no capture) we get (cid:104) (cid:105) ∀ (cid:81) − p (0, . . . , 0) = p j = 1, ..., t 1 j H1 − − − so that the fundamental probability P = (1 p ()) t (1 p (0, . . . , 0)) = (1 p )t de- 0 1 j=2 j H1 pends on a single element (ﬁrst component) of the parameter vector p . HA As previously highlighted, the procedure of ordering and scaling a generic partial capture history deﬁned in (7) is not the only way of representing the quantization of a binary sequence. Indeed, although we have argued why our choice of g(x) can be considered reasonable in some cases (also in terms of Markovian structure) it can be open to some criticism. For example, consider the following two partial capture histories each based on ﬁve capture occasions · · · · · ⇒ 1 20 + 1 21 + 1 22 + 1 23 + 0 24 15 x = (1, 1, 1, 1, 0) g(x ) = − = = 0.484 1 1 25 1 31 · · · · · ⇒ 0 20 + 0 21 + 0 22 + 0 23 + 1 24 16 x = (0, 0, 0, 0, 1) g(x ) = − = = 0.516 2 2 25 1 31 The ﬁrst partial capture history x has a total of four captures in the ﬁrst four occasions while 1 · the second one x has only one capture in the last occasion. The mapping g( ) described in 2 (7) assigns a larger impact on the conditional probabilities to x . One can ﬁnd undesirable the 2 fact that the partial capture history x having just a single capture, even though in the last 1 occasion, yields a larger value compared to a binary sequence which has 4 captures out of 5. As a possible alternative useful mapping one can consider a function based on the total number ∈ of captures occurred for each partial capture history x H. In order to obtain a potentially continuous covariate as in (7) we rescale the range in the unit interval considering as denominator the length of each capture history as follows (cid:80) (cid:26) (cid:27) z = g (x) = g (x , . . . , x ) = ljx=1 xj ∈ 0, 1 , 2 , ..., 1 (9) n n 1 lx l l l x x x ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FLEXIBLE

text: BEHAVIORAL

text: CAPTURE-RECAPTURE

text: MODELLING

text: that

text: since

text: the

text: ﬁrst

text: partition

text: I

text: can

text: be

text: equivalently

text: reduced

text: to

text: the

text: single

text: value

text: or

text: any

text: other

text: interval

text: e

text: provided

text: that

text: e

text: This

text: is

text: basically

text: due

text: to

text: the

text: discreteness

text: of

text: the

text: observable

text: range.

text: More

text: generally,

text: an

text: alternative

text: partition

text: of

text: the

text: range

text: of

text: z

text: into

text: A

text: consecutive

text: subintervals

text: I

text: e

text: I

text: (e

text: e

text: I

text: (e

text: represents

text: a

text: meaningful

text: behavioural

text: model

text: a

text: a

text: a

text: A

text: A

text: corresponding

text: to

text: the

text: regression

text: step

text: function

text: s(z)

text: logit(p

text: z

text: I

text: a

text: A.

text: Ha

text: a

text: This

text: particularly

text: ﬂexible

text: instance

text: of

text: partitioning

text: the

text: range

text: of

text: the

text: behavioural

text: covariate

text: g(x)

text: embeds

text: some

text: of

text: the

text: original

text: models

text: proposed

text: in

text: Farcomeni

text: such

text: as

text: M

text: (Supplementary

text: Web

text: Materials,

text: and

text: In

text: fact

text: looking

text: for

text: an

text: appropriate

text: number

text: and

text: location

text: of

text: the

text: partition

text: cuts

text: e

text: in

text: terms

text: of

text: step

text: functions

text: for

text: the

text: logistic

text: regression

text: can

text: readily

text: explore

text: a

text: a

text: range

text: of

text: meaningful

text: variable

text: order

text: Markov

text: chain

text: models.

text: We

text: will

text: exploit

text: this

text: approach

text: in

text: our

text: applications.

text: As

text: already

text: mentioned

text: the

text: most

text: critical

text: parameter

text: for

text: the

text: estimation

text: of

text: N

text: is

text: the

text: probability

text: P

text: as

text: in

text: Indeed

text: when

text: we

text: partition

text: the

text: set

text: of

text: conditional

text: probabilities

text: through

text: partitioning

text: the

text: quantiﬁcation

text: z

text: g(x)

text: into

text: intervals

text: I

text: I

text: we

text: have

text: that,

text: as

text: long

text: as

text: g(x)

text: I

text: e

text: A

text: (and

text: this

text: is

text: certainly

text: true

text: for

text: all

text: partial

text: capture

text: histories

text: with

text: no

text: capture)

text: we

text: get

text: p

text: p

text: j

text: t

text: j

text: so

text: that

text: the

text: fundamental

text: probability

text: P

text: p

text: t

text: p

text: p

text: )t

text: de-

text: j

text: pends

text: on

text: a

text: single

text: element

text: (ﬁrst

text: component)

text: of

text: the

text: parameter

text: vector

text: p

text: HA

text: As

text: previously

text: highlighted,

text: the

text: procedure

text: of

text: ordering

text: and

text: scaling

text: a

text: generic

text: partial

text: capture

text: history

text: deﬁned

text: in

text: is

text: not

text: the

text: only

text: way

text: of

text: representing

text: the

text: quantization

text: of

text: a

text: binary

text: sequence.

text: Indeed,

text: although

text: we

text: have

text: argued

text: why

text: our

text: choice

text: of

text: g(x)

text: can

text: be

text: considered

text: reasonable

text: in

text: some

text: cases

text: (also

text: in

text: terms

text: of

text: Markovian

text: structure)

text: it

text: can

text: be

text: open

text: to

text: some

text: criticism.

text: For

text: example,

text: consider

text: the

text: following

text: two

text: partial

text: capture

text: histories

text: each

text: based

text: on

text: ﬁve

text: capture

text: occasions

text: x

text: g(x

text: x

text: g(x

text: The

text: ﬁrst

text: partial

text: capture

text: history

text: x

text: has

text: a

text: total

text: of

text: four

text: captures

text: in

text: the

text: ﬁrst

text: four

text: occasions

text: while

text: the

text: second

text: one

text: x

text: has

text: only

text: one

text: capture

text: in

text: the

text: last

text: occasion.

text: The

text: mapping

text: g(

text: described

text: in

text: assigns

text: a

text: larger

text: impact

text: on

text: the

text: conditional

text: probabilities

text: to

text: x

text: One

text: can

text: ﬁnd

text: undesirable

text: the

text: fact

text: that

text: the

text: partial

text: capture

text: history

text: x

text: having

text: just

text: a

text: single

text: capture,

text: even

text: though

text: in

text: the

text: last

text: occasion,

text: yields

text: a

text: larger

text: value

text: compared

text: to

text: a

text: binary

text: sequence

text: which

text: has

text: captures

text: out

text: of

text: As

text: a

text: possible

text: alternative

text: useful

text: mapping

text: one

text: can

text: consider

text: a

text: function

text: based

text: on

text: the

text: total

text: number

text: of

text: captures

text: occurred

text: for

text: each

text: partial

text: capture

text: history

text: x

text: H.

text: In

text: order

text: to

text: obtain

text: a

text: potentially

text: continuous

text: covariate

text: as

text: in

text: we

text: rescale

text: the

text: range

text: in

text: the

text: unit

text: interval

text: considering

text: as

text: denominator

text: the

text: length

text: of

text: each

text: capture

text: history

text: as

text: follows

text: z

text: g

text: (x)

text: g

text: (x

text: x

text: xj

text: n

text: n

text: lx

text: l

text: l

text: l

text: x

text: x

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: EXTRACTING A BASIS WITH FIXED BLOCK INSIDE A MATRIX PIERRE YOUSSEF ABSTRACT. Given U ran n ×(cid:16)(cid:0)mPmatrix o(cid:1)f ran(cid:17)k n whose columns are denoted by (u ) (cid:54) , sev- ⊂ { j}j m eral authors have already considered the problem of ﬁnding a subset σ 1, . . . , m such that (u ) ∈ span Rn and Tr ∈ u ut −1 is minimized. In this paper, we generalize this i i σ i σ i i problem by selecting arbitrary rank matrices instead of rank one matrices. Another generaliza- tion is considering the same problem while allowing a part of the matrix to be ﬁxed. The methods of selection employed develop into algorithms. 1. INTRODUCTION q × k k Let U be an n m matrix, we see U as an operator from l2m to l2n. We denote U the k k operator norm of U while the Hilbert-Schmidt norm of U is given by U HS = Tr (UUt). k k k k The stable rank of U is given by srank(U) := U 2HS/ U 2. Note that the stable rank is always less or equal to the rank. We denote by smax and smin the largest and smallest singular value, ⊂ { } respectively. Given σ 1, ..., m , we denote U the restriction of U to the columns with indices in σ i.e. U = UP t where P : Rm −→ σRσ is the canonical coordinate projection. When σ = ∅, we haσve U =σ0. Finallyσ, if A and B are n × n symmetric matrices, the notation (cid:22) − σ A B means that B A is positive semideﬁnite. Column subset selection usualy refers to extracting from a matrix a column submatrix that has some distinguished properties. First results on column selection problems were obtained by Kashin [10]. The aim of the selection was to ﬁnd a submatrix which minimizes the operator norm among all restrictions of the same size. It was later sharpened in [12] and [9] where, × (cid:54) ⊂ { } for any n m matrix U a√nd any λ 1/4, √it is proved that there exists σ 1, . . . , m of k k (cid:54) k k k k size λm with U C( λ U + U HS/ m), with C being a universal constant. In [4], σ Bourgain-Tzafriri considered selecting a block of columns which is well invertible i.e. whose smallest singular value is bounded away from zero; their result states that there exist univer- sal constants C and C0 such that for any n × n matrix A whose columns are of norm 1, one can ﬁnd σ ⊂ {1, . . . , m} of size at least Csrank(A) such that smin(A ) (cid:62) C0. Later, Vershynin σ [16] extended the restricted invertibility principle of Bourgain-Tzafriri to the case of rectangular matrices. Moreover, he also studied the extraction of a well conditioned submatrix and proved ∈ × ⊂ { } that for any ε (0, 1) and any n m matrix U, there exists σ 1, . . . , m of size at least − (cid:54) (1 ε)srank(U) such that smax(U )/smin(U ) εclog(ε). These results were very important σ σ in geometric functional analysis and had several applications. However, the proofs were not constructive as they were based on random selection and made use of Grothendieck’s factor- ization theorem. In [15], Tropp was able to provide a randomized polynomial time algorithm to achieve the selection promised by the results of Bourgain-Tzafriri and Kashin-Tzafriri. In [14], Spielman-Srivastava produced a deterministic polynomial time algorithm to ﬁnd a well invertible submatrix inside any rectangular matrix, generalizing and improving the restricted invertibility principle of Bourgain-Tzafriri. Their proof is inspired by the method developed by Batson-Spielman-Srivastava [3] to ﬁnd a spectral sparsiﬁer of a graph. In that paper, they gave a deterministic polynomial time algorithm to ﬁnd a submatrix, with much fewer columns, Date: 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EXTRACTING

text: A

text: BASIS

text: WITH

text: FIXED

text: BLOCK

text: INSIDE

text: A

text: MATRIX

text: PIERRE

text: YOUSSEF

text: ABSTRACT.

text: Given

label: abstract

text: U

text: ran

text: n

text: n

text: whose

text: columns

text: are

text: denoted

text: by

text: (u

text: sev-

text: j}j

text: m

text: eral

text: authors

text: have

text: already

text: considered

text: the

text: problem

text: of

text: ﬁnding

text: a

text: subset

text: σ

text: m

text: such

text: that

text: (u

text: span

text: Rn

text: and

text: Tr

text: u

text: ut

text: −1

text: is

text: minimized.

text: In

text: this

text: paper,

text: we

text: generalize

text: this

text: i

text: i

text: σ

text: i

text: σ

text: i

text: i

text: problem

text: by

text: selecting

text: arbitrary

text: rank

text: matrices

text: instead

text: of

text: rank

text: one

text: matrices.

text: Another

text: generaliza-

text: tion

text: is

text: considering

text: the

text: same

text: problem

text: while

text: allowing

text: a

text: part

text: of

text: the

text: matrix

text: to

text: be

text: ﬁxed.

text: The

text: methods

text: of

text: selection

text: employed

text: develop

text: into

text: algorithms.

text: INTRODUCTION

text: q

text: k

text: k

text: Let

text: U

text: be

text: an

text: n

text: m

text: matrix,

text: we

text: see

text: U

text: as

text: an

text: operator

text: from

text: to

text: We

text: denote

text: U

text: the

text: k

text: k

text: operator

text: norm

text: of

text: U

text: while

text: the

text: Hilbert-Schmidt

text: norm

text: of

text: U

text: is

text: given

text: by

text: U

text: HS

text: Tr

text: (UUt).

text: k

text: k

text: k

text: k

text: The

text: stable

text: rank

text: of

text: U

text: is

text: given

text: by

text: srank(U)

text: U

text: U

text: Note

text: that

text: the

text: stable

text: rank

text: is

text: always

text: less

text: or

text: equal

text: to

text: the

text: rank.

text: We

text: denote

text: by

text: smax

text: and

text: smin

text: the

text: largest

text: and

text: smallest

text: singular

text: value,

text: respectively.

text: Given

text: σ

text: m

text: we

text: denote

text: U

text: the

text: restriction

text: of

text: U

text: to

text: the

text: columns

text: with

text: indices

text: in

text: σ

text: i.e.

text: U

text: UP

text: t

text: where

text: P

text: Rm

text: σRσ

text: is

text: the

text: canonical

text: coordinate

text: projection.

text: When

text: σ

text: we

text: haσve

text: U

text: Finallyσ,

text: if

text: A

text: and

text: B

text: are

text: n

text: n

text: symmetric

text: matrices,

text: the

text: notation

text: σ

text: A

text: B

text: means

text: that

text: B

text: A

text: is

text: positive

text: semideﬁnite.

text: Column

text: subset

text: selection

text: usualy

text: refers

text: to

text: extracting

text: from

text: a

text: matrix

text: a

text: column

text: submatrix

text: that

text: has

text: some

text: distinguished

text: properties.

text: First

text: results

text: on

text: column

text: selection

text: problems

text: were

text: obtained

text: by

text: Kashin

text: The

text: aim

text: of

text: the

text: selection

text: was

text: to

text: ﬁnd

text: a

text: submatrix

text: which

text: minimizes

text: the

text: operator

text: norm

text: among

text: all

text: restrictions

text: of

text: the

text: same

text: size.

text: It

text: was

text: later

text: sharpened

text: in

text: and

text: where,

text: for

text: any

text: n

text: m

text: matrix

text: U

text: a√nd

text: any

text: λ

text: √it

text: is

text: proved

text: that

text: there

text: exists

text: σ

text: m

text: of

text: k

text: k

text: k

text: k

text: k

text: k

text: size

text: λm

text: with

text: U

text: C(

text: λ

text: U

text: U

text: HS/

text: m),

text: with

text: C

text: being

text: a

text: universal

text: constant.

text: In

text: σ

text: Bourgain-Tzafriri

text: considered

text: selecting

text: a

text: block

text: of

text: columns

text: which

text: is

text: well

text: invertible

text: i.e.

text: whose

text: smallest

text: singular

text: value

text: is

text: bounded

text: away

text: from

text: zero;

text: their

text: result

text: states

text: that

text: there

text: exist

text: univer-

text: sal

text: constants

text: C

text: and

text: such

text: that

text: for

text: any

text: n

text: n

text: matrix

text: A

text: whose

text: columns

text: are

text: of

text: norm

text: one

text: can

text: ﬁnd

text: σ

text: m}

text: of

text: size

text: at

text: least

text: Csrank(A)

text: such

text: that

text: smin(A

text: Later,

text: Vershynin

text: σ

text: extended

text: the

text: restricted

text: invertibility

text: principle

text: of

text: Bourgain-Tzafriri

text: to

text: the

text: case

text: of

text: rectangular

text: matrices.

text: Moreover,

text: he

text: also

text: studied

text: the

text: extraction

text: of

text: a

text: well

text: conditioned

text: submatrix

text: and

text: proved

text: that

text: for

text: any

text: ε

text: and

text: any

text: n

text: m

text: matrix

text: U,

text: there

text: exists

text: σ

text: m

text: of

text: size

text: at

text: least

text: ε)srank(U)

text: such

text: that

text: smax(U

text: )/smin(U

text: εclog(ε).

text: These

text: results

text: were

text: very

text: important

text: σ

text: σ

text: in

text: geometric

text: functional

text: analysis

text: and

text: had

text: several

text: applications.

text: However,

text: the

text: proofs

text: were

text: not

text: constructive

text: as

text: they

text: were

text: based

text: on

text: random

text: selection

text: and

text: made

text: use

text: of

text: Grothendieck’s

text: factor-

text: ization

text: theorem.

text: In

text: Tropp

text: was

text: able

text: to

text: provide

text: a

text: randomized

text: polynomial

text: time

text: algorithm

text: to

text: achieve

text: the

text: selection

text: promised

text: by

text: the

text: results

text: of

text: Bourgain-Tzafriri

text: and

text: Kashin-Tzafriri.

text: In

text: Spielman-Srivastava

text: produced

text: a

text: deterministic

text: polynomial

text: time

text: algorithm

text: to

text: ﬁnd

text: a

text: well

text: invertible

text: submatrix

text: inside

text: any

text: rectangular

text: matrix,

text: generalizing

text: and

text: improving

text: the

text: restricted

text: invertibility

text: principle

text: of

text: Bourgain-Tzafriri.

text: Their

text: proof

text: is

text: inspired

text: by

text: the

text: method

text: developed

text: by

text: Batson-Spielman-Srivastava

text: to

text: ﬁnd

text: a

text: spectral

text: sparsiﬁer

text: of

text: a

text: graph.

text: In

text: that

text: paper,

text: they

text: gave

text: a

text: deterministic

text: polynomial

text: time

text: algorithm

text: to

text: ﬁnd

text: a

text: submatrix,

text: with

text: much

text: fewer

text: columns,

text: Date:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 P P The second generalization concerns arbitrary rank matrices; indeed, Theorem 1.1 states that ⊂ { } given UUt = i(cid:54)m uiuti, t heXre exist!s σ 1, .., m of size n(cid:16) such(cid:17)that i∈σ uiuti is invertible and −1 (cid:54) − · −1 Tr ( + 1) Tr u ut m n UUt . i i ∈ i σ Whereas Theorem 1.1 only deals with rank one matrices, we consider the same problem when replacing the rank one matrices by arbitrary rank matrices. Extending column selection prob- lems to an arbitrary rank setting is natural and can be very useful. In [5], the authors extended the spectral sparsiﬁcation of [3] to an arbitrary rank setting and showed how it can be used in order to ﬁnd a spectral sparsiﬁer of a graph while controlling additional parameters like the cost functions. The main result of this paper is the following: P × Theorem 1.2. Let A, B, B1, ..., B be n n positive semideﬁnite matrices such that A = B + (cid:54) B is of rank n. The"n fomr any k (cid:62) n − bTr (A#−1B)c, there exists σ of size k such i m(cid:16) i (cid:17) (cid:16) (cid:17) k that − + Tr ( −1 ) + 1 ( − ) · Tr ( −2 ) Tr A−σk1 (cid:54)PTr A−1 · mk − nn+ 1 + TAr (AB−1B) − k −mn +k1 + TrA(A−B1B), where A = B + ∈ B . In parσtikcular, therei eσxkistsi σ ⊂ {1, ..., m} wijth (cid:16) (cid:17)k | | = − Tr −1 σ n A B (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) such that Tr −1 (cid:54) P− + 1 + Tr −1 · Tr −1 − Tr −2 − + Tr −1 A m n A B A A B m n A B , σ where A = B + ∈ B . σ i σ i With the previous notations, B is the matrix that we would like to preserve while selecting among the (B ) (cid:54) the ones that will minimize the trace of the inverse of the corresponding i i m sum. To have a better understanding on the limitation on k in the previous statement, take A to be R the identity on n, B a projection of rank l and B1, . . . , B rank one projections. Note that in this case bTr (A−1B)c = l. To form a rank n matrix whmile keeping B, one should at least choose n − l rank one matrices which is exactly the limitation given by n − bTr (A−1B)c. In section 2, we will discuss how this result generalizes Theorem 1.1 and state direct conse- quences of it. The proof of Theorem 1.2 will be given in section 3. 2. DERIVED RESULTS Let us ﬁrst derive the rank one case of Theorem 1.2. This case is interesting since the se- lection can be traced as a selection of columns inside the matrix. Preserving the matrix B in Theorem 1.2 plays the role of preserving a block inside the matrix. Precisely, we have the following: × ⊂ { } Theorem 2.1. Let U be an n m matrix of rank n and ν 1, .., m . Denote by V the block of columns inside U corresponding to ν i.e. V = U . Let A = UUt and B = V V t. For any k (cid:62) n − bTr (A−1B)c, there exis"ts σ0 ⊂ νc of size νk such th#at if σ = σ0 ∪ ν then U is of (cid:16) (cid:17) (cid:16) (cid:17) k k k σk rank n and Tr −1 (cid:54) Tr −1 · |νc| − n + Tr (A−1B) + 1 − (|νc| − k) · Tr (A−2B) UσkUσtk A k − n + 1 + Tr (A−1B) k − n + 1 + Tr (A−1B). ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P

text: P

text: The

text: second

text: generalization

text: concerns

text: arbitrary

text: rank

text: matrices;

text: indeed,

text: Theorem

text: states

text: that

text: given

text: UUt

text: uiuti,

text: t

text: heXre

text: exist!s

text: σ

text: m

text: of

text: size

text: i∈σ

text: uiuti

text: is

text: invertible

text: and

text: −1

text: −1

text: Tr

text: Tr

text: u

text: ut

text: m

text: n

text: UUt

text: i

text: i

text: i

text: σ

text: Whereas

text: Theorem

text: only

text: deals

text: with

text: rank

text: one

text: matrices,

text: we

text: consider

text: the

text: same

text: problem

text: when

text: replacing

text: the

text: rank

text: one

text: matrices

text: by

text: arbitrary

text: rank

text: matrices.

text: Extending

text: column

text: selection

text: prob-

text: lems

text: to

text: an

text: arbitrary

text: rank

text: setting

text: is

text: natural

text: and

text: can

text: be

text: very

text: useful.

text: In

text: the

text: authors

text: extended

text: the

text: spectral

text: sparsiﬁcation

text: of

text: to

text: an

text: arbitrary

text: rank

text: setting

text: and

text: showed

text: how

text: it

text: can

text: be

text: used

text: in

text: order

text: to

text: ﬁnd

text: a

text: spectral

text: sparsiﬁer

text: of

text: a

text: graph

text: while

text: controlling

text: additional

text: parameters

text: like

text: the

text: cost

text: functions.

text: The

text: main

text: result

text: of

text: this

text: paper

text: is

text: the

text: following:

text: P

text: Theorem

text: Let

text: A,

text: B,

text: B

text: be

text: n

text: n

text: positive

text: semideﬁnite

text: matrices

text: such

text: that

text: A

text: B

text: B

text: is

text: of

text: rank

text: n.

text: The"n

text: fomr

text: any

text: k

text: n

text: bTr

text: there

text: exists

text: σ

text: of

text: size

text: k

text: such

text: i

text: i

text: k

text: that

text: Tr

text: −1

text: Tr

text: −2

text: Tr

text: mk

text: nn+

text: TAr

text: k

text: −mn

text: where

text: A

text: B

text: B

text: In

text: parσtikcular,

text: therei

text: eσxkistsi

text: σ

text: m}

text: wijth

text: Tr

text: −1

text: σ

text: n

text: A

text: B

text: such

text: that

text: Tr

text: −1

text: P−

text: Tr

text: −1

text: Tr

text: −1

text: Tr

text: −2

text: Tr

text: −1

text: A

text: m

text: n

text: A

text: B

text: A

text: A

text: B

text: m

text: n

text: A

text: B

text: σ

text: where

text: A

text: B

text: B

text: σ

text: i

text: σ

text: i

text: With

text: the

text: previous

text: notations,

text: B

text: is

text: the

text: matrix

text: that

text: we

text: would

text: like

text: to

text: preserve

text: while

text: selecting

text: among

text: the

text: (B

text: the

text: ones

text: that

text: will

text: minimize

text: the

text: trace

text: of

text: the

text: inverse

text: of

text: the

text: corresponding

text: i

text: i

text: m

text: sum.

text: To

text: have

text: a

text: better

text: understanding

text: on

text: the

text: limitation

text: on

text: k

text: in

text: the

text: previous

text: statement,

text: take

text: A

text: to

text: be

text: R

text: the

text: identity

text: on

text: n,

text: B

text: a

text: projection

text: of

text: rank

text: l

text: and

text: B

text: rank

text: one

text: projections.

text: Note

text: that

text: in

text: this

text: case

text: bTr

text: l.

text: To

text: form

text: a

text: rank

text: n

text: matrix

text: whmile

text: keeping

text: B,

text: one

text: should

text: at

text: least

text: choose

text: n

text: l

text: rank

text: one

text: matrices

text: which

text: is

text: exactly

text: the

text: limitation

text: given

text: by

text: n

text: bTr

text: In

text: section

text: we

text: will

text: discuss

text: how

text: this

text: result

text: generalizes

text: Theorem

text: and

text: state

text: direct

text: conse-

text: quences

text: of

text: it.

text: The

text: proof

text: of

text: Theorem

text: will

text: be

text: given

text: in

text: section

text: DERIVED

text: RESULTS

text: Let

text: us

text: ﬁrst

text: derive

text: the

text: rank

text: one

text: case

text: of

text: Theorem

text: This

text: case

text: is

text: interesting

text: since

text: the

text: se-

text: lection

text: can

text: be

text: traced

text: as

text: a

text: selection

text: of

text: columns

text: inside

text: the

text: matrix.

text: Preserving

text: the

text: matrix

text: B

text: in

text: Theorem

text: plays

text: the

text: role

text: of

text: preserving

text: a

text: block

text: inside

text: the

text: matrix.

text: Precisely,

text: we

text: have

text: the

text: following:

text: Theorem

text: Let

text: U

text: be

text: an

text: n

text: m

text: matrix

text: of

text: rank

text: n

text: and

text: ν

text: m

text: Denote

text: by

text: V

text: the

text: block

text: of

text: columns

text: inside

text: U

text: corresponding

text: to

text: ν

text: i.e.

text: V

text: U

text: Let

text: A

text: UUt

text: and

text: B

text: V

text: V

text: t.

text: For

text: any

text: k

text: n

text: bTr

text: there

text: exis"ts

text: σ0

text: νc

text: of

text: size

text: νk

text: such

text: th#at

text: if

text: σ

text: σ0

text: ν

text: then

text: U

text: is

text: of

text: k

text: k

text: k

text: σk

text: rank

text: n

text: and

text: Tr

text: −1

text: Tr

text: −1

text: |νc|

text: n

text: Tr

text: (|νc|

text: k)

text: Tr

text: UσkUσtk

text: A

text: k

text: n

text: Tr

text: k

text: n

text: Tr

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 PIERRE YOUSSEF j (cid:16) (cid:17)k 0 ⊂ In particular, there exists σ νc with | 0| = − Tr −1 σ n A B , (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) such that Tr −1 (cid:54) | | − + 1 + Tr −1 ·Tr −1 −Tr −2 | | − + Tr −1 U Ut νc n A B A A B νc n A B , σ σ where σ = σ0 ∪ ν. P P ∈ Proof. Denote by (u ) (cid:54) the columns of U. Clearly, UUt = (cid:54) u ut. For any i ν, deﬁne B = u ut. Then weihiavme A = B + ∈ B . Theorem 2.1 folliowms bi yi applying Theorem 1.2 toiA, B,i (Bi ) ∈ with m replaced by |νic|.νc i (cid:3) i i νc Theorem 2.1 allows one to preserve the block V while achieving the desired extraction. Let us illustrate how this can be useful. For this aim, consider the case where UUt = Id. × Corollary 2.2. Let U be an n m matrix such that UUt = Id. Suppose that U contains r (cid:54) ⊂ { } columns of norm 1 for some r n. Then there exists σ 1, .., m of size n such that U is σ invertible and k −1k2 (cid:54) ( − ) · ( − ) + P U HS m n n r n. σ Proof. First note that r = n implies that m = n and the problem is trivial. Indeed, if n < m k k we have n = Tr (UUt) = i(cid:54)m uj 22 > r, where (uj)j(cid:54)m denote the columns of U. This of course contradicts the fact that r = n. Now let us consider the case r < n. Suppose that UUt = Id and that U contains r columns ⊂ { } of norm 1. Let ν 1, .., m be the set of indices of the norm one columns and denote V = U . × k k (cid:54) k k · k k (cid:54) ν Clearly, V is an n r matrix of rank r since r = V 2HS V 2 rank(V ) and V 1. By preserving V , we already have r linearly independent columns and therefore we just need − to complement them by n r linearly independent columns. Following the notations of Theorem 2.j1, w(cid:16)e have(cid:17)Ak = Id and B = V V t. Applying Theo- 0 rem 2.1, one can ﬁnd σ with | 0| = − Tr −1 = − σ n A B n r, (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) such that if σ = σ0 ∪ ν, then |σ| = n and k −1k2 (cid:54) | | − + 1 + Tr −1 · Tr −1 − Tr −2 | | − + Tr −1 U HS νc n A B A A B νc n A B σ − − · − · − − = ( + 1 + ) ( + ) m r n r n r m r n r − · − = ( ) ( ) + m n n r n. (cid:3) Remark 2.3. When we don’t look for columns of norm 1 inside U (i.e. the case where r = 0), 6 the estimate would be the same as in Theorem 1.1 while if r = 0, the estimate of Corollary 2.2 · − improves the one of Theorem 1.1 by r (m n). Moreover, one can view Corollary 2.2 as an extraction of a basis inside U with a "good" estimate on the Hilbert-Schmidt of the inverse while preserving a "nice" block. To have a better understanding of the generalization of Theorem 1.1 to the case of arbitrary rPank matrices, let us take B = 0 in Theorem 1.2 to get the following corollary: × CPorollary 2.4. Let A, B1, . . . , B be n n positive semideﬁnite matrices such that A = m (cid:62) ⊂ { } (cid:54) B is of rank n. Then for any k n, there exists σ 1, .., m of size k such that i m i ! ∈ B is invertible and X (cid:16) (cid:17) i σ i −1 − + 1 Tr (cid:54) m n Tr −1 B − A . i + 1 ∈ k n i σ ##LTLine##

text: PIERRE

text: YOUSSEF

text: j

text: In

text: particular,

text: there

text: exists

text: σ

text: νc

text: with

text: Tr

text: −1

text: σ

text: n

text: A

text: B

text: such

text: that

text: Tr

text: −1

text: Tr

text: −1

text: ·Tr

text: −1

text: −Tr

text: −2

text: Tr

text: −1

text: U

text: Ut

text: νc

text: n

text: A

text: B

text: A

text: A

text: B

text: νc

text: n

text: A

text: B

text: σ

text: σ

text: where

text: σ

text: σ0

text: ν.

text: P

text: P

text: Proof.

text: Denote

text: by

text: (u

text: the

text: columns

text: of

text: U.

text: Clearly,

text: UUt

text: u

text: ut.

text: For

text: any

text: i

text: ν,

text: deﬁne

text: B

text: u

text: ut.

text: Then

text: weihiavme

text: A

text: B

text: B

text: Theorem

text: folliowms

text: bi

text: yi

text: applying

text: Theorem

text: toiA,

text: B,i

text: (Bi

text: with

text: m

text: replaced

text: by

text: |νic|.νc

text: i

text: i

text: i

text: νc

text: Theorem

text: allows

text: one

text: to

text: preserve

text: the

text: block

text: V

text: while

text: achieving

text: the

text: desired

text: extraction.

text: Let

text: us

text: illustrate

text: how

text: this

text: can

text: be

text: useful.

text: For

text: this

text: aim,

text: consider

text: the

text: case

text: where

text: UUt

text: Id.

text: Corollary

text: Let

text: U

text: be

text: an

text: n

text: m

text: matrix

text: such

text: that

text: UUt

text: Id.

text: Suppose

text: that

text: U

text: contains

text: r

text: columns

text: of

text: norm

text: for

text: some

text: r

text: n.

text: Then

text: there

text: exists

text: σ

text: m

text: of

text: size

text: n

text: such

text: that

text: U

text: is

text: σ

text: invertible

text: and

text: k

text: P

text: U

text: HS

text: m

text: n

text: n

text: r

text: n.

text: σ

text: Proof.

text: First

text: note

text: that

text: r

text: n

text: implies

text: that

text: m

text: n

text: and

text: the

text: problem

text: is

text: trivial.

text: Indeed,

text: if

text: n

text: m

text: k

text: k

text: we

text: have

text: n

text: Tr

text: (UUt)

text: uj

text: r,

text: where

text: denote

text: the

text: columns

text: of

text: U.

text: This

text: of

text: course

text: contradicts

text: the

text: fact

text: that

text: r

text: n.

text: Now

text: let

text: us

text: consider

text: the

text: case

text: r

text: n.

text: Suppose

text: that

text: UUt

text: Id

text: and

text: that

text: U

text: contains

text: r

text: columns

text: of

text: norm

text: Let

text: ν

text: m

text: be

text: the

text: set

text: of

text: indices

text: of

text: the

text: norm

text: one

text: columns

text: and

text: denote

text: V

text: U

text: k

text: k

text: k

text: k

text: k

text: k

text: ν

text: Clearly,

text: V

text: is

text: an

text: n

text: r

text: matrix

text: of

text: rank

text: r

text: since

text: r

text: V

text: V

text: rank(V

text: and

text: V

text: By

text: preserving

text: V

text: we

text: already

text: have

text: r

text: linearly

text: independent

text: columns

text: and

text: therefore

text: we

text: just

text: need

text: to

text: complement

text: them

text: by

text: n

text: r

text: linearly

text: independent

text: columns.

text: Following

text: the

text: notations

text: of

text: Theorem

text: Id

text: and

text: B

text: V

text: V

text: t.

text: Applying

text: Theo-

text: rem

text: one

text: can

text: ﬁnd

text: σ

text: with

text: Tr

text: −1

text: σ

text: n

text: A

text: B

text: n

text: r,

text: such

text: that

text: if

text: σ

text: σ0

text: ν,

text: then

text: |σ|

text: n

text: and

text: k

text: Tr

text: −1

text: Tr

text: −1

text: Tr

text: −2

text: Tr

text: −1

text: U

text: HS

text: νc

text: n

text: A

text: B

text: A

text: A

text: B

text: νc

text: n

text: A

text: B

text: σ

text: m

text: r

text: n

text: r

text: n

text: r

text: m

text: r

text: n

text: r

text: m

text: n

text: n

text: r

text: n.

text: Remark

text: When

text: we

text: don’t

text: look

text: for

text: columns

text: of

text: norm

text: inside

text: U

text: (i.e.

text: the

text: case

text: where

text: r

text: the

text: estimate

text: would

text: be

text: the

text: same

text: as

text: in

text: Theorem

text: while

text: if

text: r

text: the

text: estimate

text: of

text: Corollary

text: improves

text: the

text: one

text: of

text: Theorem

text: by

text: r

text: (m

text: n).

text: Moreover,

text: one

text: can

text: view

text: Corollary

text: as

text: an

text: extraction

text: of

text: a

text: basis

text: inside

text: U

text: with

text: a

text: "good"

text: estimate

text: on

text: the

text: Hilbert-Schmidt

text: of

text: the

text: inverse

text: while

text: preserving

text: a

text: "nice"

text: block.

text: To

text: have

text: a

text: better

text: understanding

text: of

text: the

text: generalization

text: of

text: Theorem

text: to

text: the

text: case

text: of

text: arbitrary

text: rPank

text: matrices,

text: let

text: us

text: take

text: B

text: in

text: Theorem

text: to

text: get

text: the

text: following

text: corollary:

text: CPorollary

text: Let

text: A,

text: B

text: be

text: n

text: n

text: positive

text: semideﬁnite

text: matrices

text: such

text: that

text: A

text: m

text: B

text: is

text: of

text: rank

text: n.

text: Then

text: for

text: any

text: k

text: n,

text: there

text: exists

text: σ

text: m

text: of

text: size

text: k

text: such

text: that

text: i

text: m

text: i

text: B

text: is

text: invertible

text: and

text: X

text: i

text: σ

text: i

text: −1

text: Tr

text: m

text: n

text: Tr

text: −1

text: B

text: A

text: i

text: k

text: n

text: i

text: σ

text: ##LTLine##


text: 5 3. PROOF OF THEOREM 1.2 ThePproof is based on an iteration of the following lemma: × Lemma 3.1. Let A , B, B1, ..., B be n n positive semideﬁnite matrices such that A = p p ∈ { } − p B + (cid:54) B is of rank n. Then there exists j 1, .., p such that A B is of rank n satisfyinigp(Aip − Bj)−1 (cid:23) A(cid:16)−p 1 an(cid:17)d  (cid:16) (cid:17)  p(cid:16) j (cid:17)  (cid:16) (cid:17)  (cid:16) (cid:17) − + Tr −1 + 1 Tr −2 Tr ( − )−1 (cid:54) Tr −1 · p n Ap B − Ap B Ap Bj Ap − + Tr −1 − + Tr −1 . p n A B p n A B p p − Proof. Our aim is to ﬁnd C among the (B ) (cid:54) such that A C is still invertible and then have a control on Tr (A − C)−1. We would iliki ep to use the Spherman-Morrison-Woodbury which p states that (cid:16) (cid:17) (1) (A − C)−1 = A−1 + A−1C 12 Id − C 12 A−1C 12 −1 C 12 A−1. p p p p p (cid:13) (cid:13) SinFcoerCth12isAf−or1mCu12laistoa hpoolsdit,iwveedsehﬁonuiltdeemnsautrriexc, hitoiosssinugfﬁCciesun(cid:13)(cid:13)(cid:13)ct htothhaatvIed(cid:13)(cid:13)−C(cid:13)(cid:13)(cid:13)C12 A12 A−1−pC1C(cid:16)12 (cid:13)(cid:13)12 <is i1n(cid:17)vinerotirbdleer. p (cid:16) (cid:17) p to ensure the invertibility of Id − C 12 A−1C 12 . Now since C 12 A−1C 12 (cid:54) Tr A−1C then it (cid:16) (cid:17) p p p would be sufﬁcient to have 1 − Tr A−1C > 0 in order to use (1). p Since Tr A−2C is positive t(cid:16)hen we(cid:17)may se(cid:16)arch for(cid:16)C satis(cid:17)f(cid:17)ying p (2) Tr A−2C (cid:54) α 1 − Tr A−1C , p p where α is a positive parameter which will be chosen later. In order to guarantee the existence of C among the (B ) (cid:54) that satisﬁes (2), it is sufﬁcient to i i p prove that (2) holds when taking the sum overall (B ) (cid:54) . Therefore, we need to prove that i i p X (cid:16) (cid:17) X (cid:16) (cid:17)   (3) Tr A−2B (cid:54) α p − Tr A−1B . (cid:54) p P i (cid:54) p i i p i p − Now since the trace is(cid:16) linea(cid:17)r and (cid:16) i(cid:54)p Bi(cid:17)= Ap (cid:16) B then (3)(cid:16)is equiv(cid:17)a(cid:17)lent to the following (4) Tr A−1 − Tr A−2B (cid:54) α p − n + Tr A−1B . p p p Choose α so that (4) becomes true. For that let α = Tr(A−−p +1)T−r(Tr(−A1−p 2)B). Therefore, there exists j (cid:54) p such that C = B satisﬁeps (n2). NAopwB, we may use the Sherman- (cid:18)j (cid:19) Morrison-Woodbury formula (1) and write −1 ( − )−1 = −1 + −1 1 − 1 −1 1 1 −1 A B A A B 2 Id B 2 A B 2 B 2 A p j p p j (cid:16) (cid:17)j p j j p −1 −1 (cid:22) A−p 1 + 1 −ATp rBjA−p1 . A B p j Now since B satisﬁes (2), then j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: PROOF

text: OF

text: THEOREM

text: ThePproof

text: is

text: based

text: on

text: an

text: iteration

text: of

text: the

text: following

text: lemma:

text: Lemma

text: Let

text: A

text: B,

text: B

text: be

text: n

text: n

text: positive

text: semideﬁnite

text: matrices

text: such

text: that

text: A

text: p

text: p

text: p

text: B

text: B

text: is

text: of

text: rank

text: n.

text: Then

text: there

text: exists

text: j

text: p

text: such

text: that

text: A

text: B

text: is

text: of

text: rank

text: n

text: satisfyinigp(Aip

text: j

text: Tr

text: −1

text: Tr

text: −2

text: Tr

text: Tr

text: −1

text: p

text: n

text: Ap

text: B

text: Ap

text: B

text: Ap

text: Bj

text: Ap

text: Tr

text: −1

text: Tr

text: −1

text: p

text: n

text: A

text: B

text: p

text: n

text: A

text: B

text: p

text: p

text: Proof.

text: Our

text: aim

text: is

text: to

text: ﬁnd

text: C

text: among

text: the

text: (B

text: such

text: that

text: A

text: C

text: is

text: still

text: invertible

text: and

text: then

text: have

text: a

text: control

text: on

text: Tr

text: (A

text: We

text: would

text: iliki

text: ep

text: to

text: use

text: the

text: Spherman-Morrison-Woodbury

text: which

text: p

text: states

text: that

text: (A

text: Id

text: C

text: −1

text: C

text: p

text: p

text: p

text: p

text: p

text: <is

text: p

text: p

text: to

text: ensure

text: the

text: invertibility

text: of

text: Id

text: C

text: Now

text: since

text: C

text: Tr

text: then

text: it

text: p

text: p

text: p

text: would

text: be

text: sufﬁcient

text: to

text: have

text: Tr

text: in

text: order

text: to

text: use

text: p

text: Since

text: Tr

text: is

text: positive

text: p

text: Tr

text: α

text: Tr

text: p

text: p

text: where

text: α

text: is

text: a

text: positive

text: parameter

text: which

text: will

text: be

text: chosen

text: later.

text: In

text: order

text: to

text: guarantee

text: the

text: existence

text: of

text: C

text: among

text: the

text: (B

text: that

text: satisﬁes

text: it

text: is

text: sufﬁcient

text: to

text: i

text: i

text: p

text: prove

text: that

text: holds

text: when

text: taking

text: the

text: sum

text: overall

text: (B

text: Therefore,

text: we

text: need

text: to

text: prove

text: that

text: i

text: i

text: p

text: X

text: X

text: Tr

text: α

text: p

text: Tr

text: p

text: P

text: i

text: p

text: i

text: i

text: p

text: i

text: p

text: Now

text: since

text: the

text: trace

text: and

text: Ap

text: B

text: then

text: to

text: the

text: following

text: Tr

text: Tr

text: α

text: p

text: n

text: Tr

text: p

text: p

text: p

text: Choose

text: α

text: so

text: that

text: becomes

text: true.

text: For

text: that

text: let

text: α

text: Tr(A−−p

text: Therefore,

text: there

text: exists

text: j

text: p

text: such

text: that

text: C

text: B

text: satisﬁeps

text: NAopwB,

text: we

text: may

text: use

text: the

text: Sherman-

text: Morrison-Woodbury

text: formula

text: and

text: write

text: −1

text: −1

text: −1

text: −1

text: −1

text: A

text: B

text: A

text: A

text: B

text: Id

text: B

text: A

text: B

text: B

text: A

text: p

text: j

text: p

text: p

text: j

text: p

text: j

text: j

text: p

text: −1

text: −1

text: A−p

text: −ATp

text: A

text: B

text: p

text: j

text: Now

text: since

text: B

text: satisﬁes

text: then

text: j

text: ##LTLine##

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