arXiv subjects

Manolis C. Tsakiris

Publications and source records attributed to Manolis C. Tsakiris.

33 records · Page 2Linked to original sources

arXiv2021

Boosting RANSAC via Dual Principal Component Pursuit

In this paper, we revisit the problem of local optimization in RANSAC. Once a so-far-the-best model has been found, we refine it via Dual Principal Component Pursuit (DPCP), a robust subspace learning method with strong theoretical support and efficient algorithms. The proposed DPCP-RANSAC has far fewer parameters than existing methods and is scalable. Experiments on estimating two-view homographies, fundamental and essential matrices, and three-view homographic tensors using large-scale datasets show that our approach consistently has higher accuracy than state-of-the-art alternatives.

cs.CV↗

arXiv2020

Determinantal conditions for homomorphic sensing

With $k$ an infinite field and $τ_1,τ_2$ endomorphisms of $k^m$, we provide a dimension bound on an open locus of a determinantal scheme, under which, for a general subspace $V \subseteq k^m$ of dimension $n \le m/2$, for $v_1,v_2 \in V$ we have $τ_1(v_1)=τ_2(v_2)$ only if $v_1=v_2$. Specializing to permutations composed by coordinate projections, we obtain an abstract proof of the unlabeled sensing theorem.

math.CO↗

arXiv2020

Resolution of ideals associated to subspace arrangements

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.

math.AC↗

arXiv2020

Linear Regression without Correspondences via Concave Minimization

Linear regression without correspondences concerns the recovery of a signal in the linear regression setting, where the correspondences between the observations and the linear functionals are unknown. The associated maximum likelihood function is NP-hard to compute when the signal has dimension larger than one. To optimize this objective function we reformulate it as a concave minimization problem, which we solve via branch-and-bound. This is supported by a computable search space to branch, an effective lower bounding scheme via convex envelope minimization and a refined upper bound, all naturally arising from the concave minimization reformulation. The resulting algorithm outperforms state-of-the-art methods for fully shuffled data and remains tractable for up to $8$-dimensional signals, an untouched regime in prior work.

cs.IT↗

arXiv2020

Finding the Sparsest Vectors in a Subspace: Theory, Algorithms, and Applications

The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine learning. However, in contrast to the classical sparse recovery problem, the most natural formulation for finding the sparsest vector in a subspace is usually nonconvex. In this paper, we overview recent advances on global nonconvex optimization theory for solving this problem, ranging from geometric analysis of its optimization landscapes, to efficient optimization algorithms for solving the associated nonconvex optimization problem, to applications in machine intelligence, representation learning, and imaging sciences. Finally, we conclude this review by pointing out several interesting open problems for future research.

cs.LG↗

arXiv2019

Dual Principal Component Pursuit

We consider the problem of learning a linear subspace from data corrupted by outliers. Classical approaches are typically designed for the case in which the subspace dimension is small relative to the ambient dimension. Our approach works with a dual representation of the subspace and hence aims to find its orthogonal complement; as such, it is particularly suitable for subspaces whose dimension is close to the ambient dimension (subspaces of high relative dimension). We pose the problem of computing normal vectors to the inlier subspace as a non-convex $\ell_1$ minimization problem on the sphere, which we call Dual Principal Component Pursuit (DPCP) problem. We provide theoretical guarantees under which every global solution to DPCP is a vector in the orthogonal complement of the inlier subspace. Moreover, we relax the non-convex DPCP problem to a recursion of linear programs whose solutions are shown to converge in a finite number of steps to a vector orthogonal to the subspace. In particular, when the inlier subspace is a hyperplane, the solutions to the recursion of linear programs converge to the global minimum of the non-convex DPCP problem in a finite number of steps. We also propose algorithms based on alternating minimization and iteratively re-weighted least squares, which are suitable for dealing with large-scale data. Experiments on synthetic data show that the proposed methods are able to handle more outliers and higher relative dimensions than current state-of-the-art methods, while experiments in the context of the three-view geometry problem in computer vision suggest that the proposed methods can be a useful or even superior alternative to traditional RANSAC-based approaches for computer vision and other applications.

cs.CV↗

arXiv2019

An algebraic-geometric approach for linear regression without correspondences

Linear regression without correspondences is the problem of performing a linear regression fit to a dataset for which the correspondences between the independent samples and the observations are unknown. Such a problem naturally arises in diverse domains such as computer vision, data mining, communications and biology. In its simplest form, it is tantamount to solving a linear system of equations, for which the entries of the right hand side vector have been permuted. This type of data corruption renders the linear regression task considerably harder, even in the absence of other corruptions, such as noise, outliers or missing entries. Existing methods are either applicable only to noiseless data or they are very sensitive to initialization or they work only for partially shuffled data. In this paper we address these issues via an algebraic geometric approach, which uses symmetric polynomials to extract permutation-invariant constraints that the parameters $ξ^* \in \Re^n$ of the linear regression model must satisfy. This naturally leads to a polynomial system of $n$ equations in $n$ unknowns, which contains $ξ^*$ in its root locus. Using the machinery of algebraic geometry we prove that as long as the independent samples are generic, this polynomial system is always consistent with at most $n!$ complex roots, regardless of any type of corruption inflicted on the observations. The algorithmic implication of this fact is that one can always solve this polynomial system and use its most suitable root as initialization to the Expectation Maximization algorithm. To the best of our knowledge, the resulting method is the first working solution for small values of $n$ able to handle thousands of fully shuffled noisy observations in milliseconds.

cs.LG↗

arXiv2019

Homomorphic Sensing

A recent line of research termed unlabeled sensing and shuffled linear regression has been exploring under great generality the recovery of signals from subsampled and permuted measurements; a challenging problem in diverse fields of data science and machine learning. In this paper we introduce an abstraction of this problem which we call homomorphic sensing. Given a linear subspace and a finite set of linear transformations we develop an algebraic theory which establishes conditions guaranteeing that points in the subspace are uniquely determined from their homomorphic image under some transformation in the set. As a special case, we recover known conditions for unlabeled sensing, as well as new results and extensions. On the algorithmic level we exhibit two dynamic programming based algorithms, which to the best of our knowledge are the first working solutions for the unlabeled sensing problem for small dimensions. One of them, additionally based on branch-and-bound, when applied to image registration under affine transformations, performs on par with or outperforms state-of-the-art methods on benchmark datasets.

cs.IT↗

arXiv2018

Dual Principal Component Pursuit: Probability Analysis and Efficient Algorithms

Recent methods for learning a linear subspace from data corrupted by outliers are based on convex $\ell_1$ and nuclear norm optimization and require the dimension of the subspace and the number of outliers to be sufficiently small. In sharp contrast, the recently proposed Dual Principal Component Pursuit (DPCP) method can provably handle subspaces of high dimension by solving a non-convex $\ell_1$ optimization problem on the sphere. However, its geometric analysis is based on quantities that are difficult to interpret and are not amenable to statistical analysis. In this paper we provide a refined geometric analysis and a new statistical analysis that show that DPCP can tolerate as many outliers as the square of the number of inliers, thus improving upon other provably correct robust PCA methods. We also propose a scalable Projected Sub-Gradient Method method (DPCP-PSGM) for solving the DPCP problem and show it admits linear convergence even though the underlying optimization problem is non-convex and non-smooth. Experiments on road plane detection from 3D point cloud data demonstrate that DPCP-PSGM can be more efficient than the traditional RANSAC algorithm, which is one of the most popular methods for such computer vision applications.

cs.LG↗

arXiv2018

Theoretical Analysis of Sparse Subspace Clustering with Missing Entries

Sparse Subspace Clustering (SSC) is a popular unsupervised machine learning method for clustering data lying close to an unknown union of low-dimensional linear subspaces; a problem with numerous applications in pattern recognition and computer vision. Even though the behavior of SSC for complete data is by now well-understood, little is known about its theoretical properties when applied to data with missing entries. In this paper we give theoretical guarantees for SSC with incomplete data, and analytically establish that projecting the zero-filled data onto the observation pattern of the point being expressed leads to a substantial improvement in performance. The main insight that stems from our analysis is that even though the projection induces additional missing entries, this is counterbalanced by the fact that the projected and zero-filled data are in effect incomplete points associated with the union of the corresponding projected subspaces, with respect to which the point being expressed is complete. The significance of this phenomenon potentially extends to the entire class of self-expressive methods.

cs.LG↗

arXiv2017

Hyperplane Clustering Via Dual Principal Component Pursuit

We extend the theoretical analysis of a recently proposed single subspace learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case where the data are drawn from of a union of hyperplanes. To gain insight into the properties of the $\ell_1$ non-convex problem associated with DPCP, we develop a geometric analysis of a closely related continuous optimization problem. Then transferring this analysis to the discrete problem, our results state that as long as the hyperplanes are sufficiently separated, the dominant hyperplane is sufficiently dominant and the points are uniformly distributed inside the associated hyperplanes, then the non-convex DPCP problem has a unique global solution, equal to the normal vector of the dominant hyperplane. This suggests the correctness of a sequential hyperplane learning algorithm based on DPCP. A thorough experimental evaluation reveals that hyperplane learning schemes based on DPCP dramatically improve over the state-of-the-art methods for the case of synthetic data, while are competitive to the state-of-the-art in the case of 3D plane clustering for Kinect data.

cs.CV↗

arXiv2017

Filtrated Algebraic Subspace Clustering

Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In the abstract form of the problem, where no noise or other corruptions are present, the data are assumed to lie in general position inside the algebraic variety of a union of subspaces, and the objective is to decompose the variety into its constituent subspaces. Prior algebraic-geometric approaches to this problem require the subspaces to be of equal dimension, or the number of subspaces to be known. Subspaces of arbitrary dimensions can still be recovered in closed form, in terms of all homogeneous polynomials of degree $m$ that vanish on their union, when an upper bound m on the number of the subspaces is given. In this paper, we propose an alternative, provably correct, algorithm for addressing a union of at most $m$ arbitrary-dimensional subspaces, based on the idea of descending filtrations of subspace arrangements. Our algorithm uses the gradient of a vanishing polynomial at a point in the variety to find a hyperplane containing the subspace S passing through that point. By intersecting the variety with this hyperplane, we obtain a subvariety that contains S, and recursively applying the procedure until no non-trivial vanishing polynomial exists, our algorithm eventually identifies S. By repeating this procedure for other points, our algorithm eventually identifies all the subspaces by returning a basis for their orthogonal complement. Finally, we develop a variant of the abstract algorithm, suitable for computations with noisy data. We show by experiments on synthetic and real data that the proposed algorithm outperforms state-of-the-art methods on several occasions, thus demonstrating the merit of the idea of filtrations.

cs.CV↗

arXiv2017

Algebraic Clustering of Affine Subspaces

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick, which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subpaces.

cs.CV↗

arXiv2015

Filtrated Spectral Algebraic Subspace Clustering

Algebraic Subspace Clustering (ASC) is a simple and elegant method based on polynomial fitting and differentiation for clustering noiseless data drawn from an arbitrary union of subspaces. In practice, however, ASC is limited to equi-dimensional subspaces because the estimation of the subspace dimension via algebraic methods is sensitive to noise. This paper proposes a new ASC algorithm that can handle noisy data drawn from subspaces of arbitrary dimensions. The key ideas are (1) to construct, at each point, a decreasing sequence of subspaces containing the subspace passing through that point; (2) to use the distances from any other point to each subspace in the sequence to construct a subspace clustering affinity, which is superior to alternative affinities both in theory and in practice. Experiments on the Hopkins 155 dataset demonstrate the superiority of the proposed method with respect to sparse and low rank subspace clustering methods.

cs.CV↗

arXiv2014

Algebraic Decompositions of DP Problems with Linear Dynamics

Inspired by rational canonical forms, we introduce and analyze two decompositions of dynamic programming (DP) problems for systems with linear dynamics. Specifically, we consider both finite and infinite horizon DP problems in which the dynamics are linear, the cost function depends only on the state, and the state-space is finite dimensional but defined over an arbitrary algebraic field. Starting from the natural decomposition of the state-space into the direct sum of subspaces that are invariant under the system's linear transformation, and assuming that the cost functions exhibit an additive structure compatible with this decomposition, we extract from the original DP problem two distinct families of smaller DP problems, each associated with a system evolving on an invariant subspace of the original state-space. We propose that each of these families constitutes a decomposition of the original problem when the optimal policy and value function of the original problem can be reconstructed from the optimal policies and value functions of the individual subproblems in the family. We derive necessary and sufficient conditions for these decompositions to exist both in the finite and infinite horizon cases. We also propose a readily verifiable sufficient condition under which the first decomposition exists, and we show that the first notion of decomposition is generally stronger than the second.

math.OC↗