Schubert Varieties and Distances between Subspaces of Different Dimensions

Ye, Ke; Lim, Lek-Heng

doi:10.1137/15m1054201

Cited by 143 publications

(130 citation statements)

References 44 publications

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“…This bound on SE still allows the chordal distance between the two subspaces to be O(1). Chordal distance [20] is the l 2 norm of the vector containing the sine of all principal angles. The second related extra requirement is an upper bound on ∆ (slow subspace change) which depends on the value of x min .…”

Section: Discussionmentioning

confidence: 99%

Nearly Optimal Robust Subspace Tracking: A Unified Approach

Narayanamurthy

Vaswani

2018

2018 IEEE Data Science Workshop (DSW)

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In this work, we study the robust subspace tracking (RST) problem and obtain one of the first two provable guarantees for it. The goal of RST is to track sequentially arriving data vectors that lie in a slowly changing low-dimensional subspace, while being robust to corruption by additive sparse outliers. It can also be interpreted as a dynamic (time-varying) extension of robust PCA (RPCA), with the minor difference that RST also requires a short tracking delay. We develop a recursive projected compressive sensing algorithm that we call Nearly Optimal RST via ReProCS (ReProCS-NORST) because its tracking delay is nearly optimal. We prove that NORST solves both the RST and the dynamic RPCA problems under weakened standard RPCA assumptions, two simple extra assumptions (slow subspace change and most outlier magnitudes lower bounded), and a few minor assumptions.Our guarantee shows that NORST enjoys a near optimal tracking delay of O(r log n log(1/ )). Its required delay between subspace change times is the same, and its memory complexity is n times this value. Thus both these are also nearly optimal. Here n is the ambient space dimension, r is the subspaces' dimension, and is the tracking accuracy. NORST also has the best outlier tolerance compared with all previous RPCA or RST methods, both theoretically and empirically (including for real videos), without requiring any model on how the outlier support is generated. This is possible because of the extra assumptions it uses. * A shorter version of this manuscript [1] will be presented at ICML, 2018. Another small part, Corollary 5.18, will appear in [2]. arXiv:1712.06061v4 [cs.IT] 6 Jul 2018 Definition 1.1. An n × r basis matrix P is µ-incoherent if max i=1,2,..,n P (i) 2 2 ≤ µr/n. Here µ is called the coherence parameter. It quantifies the non-denseness of P .A simple way to ensure that X is not low-rank is by imposing upper bounds on max-outlier-frac-row and max-outlier-frac-col [8,9]. One way to ensure identifiability of the changing subspaces is to assume that they are piecewise constant:and to lower bound t j+1 − t j . Let t 0 = 1 and t J+1 = d. With this model, r L = rJ in general (except if subspace directions are repeated). The union of the column spans of all the P j 's is equal to the span of the left singular vectors of L. Thus, assuming that the P j 's are µ-incoherent implies their incoherence. We also assume that the subspace coefficients a t are mutually independent over time, have identical and diagonal covariance matrices denoted by Λ, and are element-wise bounded. Element-wise bounded-ness of a t 's, along with the statistical assumptions, is similar to incoherence of right singular vectors of L (right incoherence); see Remark 3.5. Because tracking requires an online algorithm that processes data vectors one at a time or in mini-batches, we need these statistical assumptions on the a t 's. For the same reason, we also need to re-define max-outlier-frac-row as the maximum fraction of nonzeroes in any row of any α-consecutive-column sub-matrix of X. ...

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Section: Discussionmentioning

confidence: 99%

Nearly Optimal Robust Subspace Tracking: A Unified Approach

Narayanamurthy

Vaswani

2018

2018 IEEE Data Science Workshop (DSW)

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“…where, r 1 and r 2 denote the ranks of the basis sets B 1 and B 2 respectively. Note, this expression is equivalent to the chordal metric [18] and is applicable to the case when r 1 = r 2 .…”

Section: Preliminariesmentioning

confidence: 99%

Multiple Subspace Alignment Improves Domain Adaptation

Thopalli

Thiagarajan

Turaga

2019

ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We present a novel unsupervised domain adaptation (DA) method for cross-domain visual recognition. Though subspace methods have found success in DA, their performance is often limited due to the assumption of approximating an entire dataset using a single low-dimensional subspace. Instead, we develop a method to effectively represent the source and target datasets via a collection of low-dimensional subspaces, and subsequently align them by exploiting the natural geometry of the space of subspaces, on the Grassmann manifold. We demonstrate the effectiveness of this approach, using empirical studies on two widely used benchmarks, with state of the art domain adaptation performance.

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“…Our method of defining a geometric distance δ + 2 for pairs of positive definite matrices of different dimensions is inspired by a similar (at least in spirit) extension of the distance on a Grassmannian to subspaces of different dimensions proposed in [14]. The following convex sets will play the role of the Schubert varieties in [14]. Given A ∈ S m ++ and B ∈ S n ++ , a natural way to define the distance between A and B is to define it as the distance from A to the set Ω − (B), i.e.,…”

Section: Geometric Distance Between Ellipsoids Of Different Dimensionsmentioning

confidence: 99%

Geometric Distance Between Positive Definite Matrices of Different Dimensions

Lim

Sepulchre

Yao

2019

IEEE Trans. Inform. Theory

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We show how the Riemannian distance on S n ++ , the cone of n × n real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that S n ++ also parameterizes n-dimensional ellipsoids, and inner products on R n , n × n covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.2010 Mathematics Subject Classification. 15B48, 15A18, 51K99, 53C25.

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Schubert Varieties and Distances between Subspaces of Different Dimensions

Cited by 143 publications

References 44 publications

Nearly Optimal Robust Subspace Tracking: A Unified Approach

Nearly Optimal Robust Subspace Tracking: A Unified Approach

Multiple Subspace Alignment Improves Domain Adaptation

Geometric Distance Between Positive Definite Matrices of Different Dimensions

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