Locality Regularized Reconstruction: Structured Sparsity and Delaunay Triangulations

Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points $[\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n] = \mathbf{X} \in \mathbb{R}^{d \times n}$ and a vector $\mathbf{y} \in \mathbb{R}^d$, the goal is to find coefficients $\mathbf{w} \in \mathbb{R}^n$ so that $\mathbf{X} \mathbf{w} \approx \mathbf{y}$, subject to some desired structure on $\mathbf{w}$. In this work we seek $\mathbf{w}$ that forms a local reconstruction of $\mathbf{y}$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $\mathbf{X}$ that are close to $\mathbf{y}$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $\mathbf{X}$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$, thereby providing local sparse solutions when $d \ll n$. Under the same condition we also show that for any $\mathbf{y}$ contained in the convex hull of $\mathbf{X}$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $\mathbf{y}$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $\mathbf{X}$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex.

A Nyström method with missing distances

We study the problem of determining the configuration of $n$ points, referred to as mobile nodes, by utilizing pairwise distances to $m$ fixed points known as anchor nodes. In the standard setting, we have information about the distances between anchors (anchor-anchor) and between anchors and mobile nodes (anchor-mobile), but the distances between mobile nodes (mobile-mobile) are not known. For this setup, the Nystr\"om method is a viable technique for estimating the positions of the mobile nodes. This study focuses on the setting where the anchor-mobile block of the distance matrix contains only partial distance information. First, we establish a relationship between the columns of the anchor-mobile block in the distance matrix and the columns of the corresponding block in the Gram matrix via a graph Laplacian. Exploiting this connection, we introduce a novel sampling model that frames the position estimation problem as low-rank recovery of an inner product matrix, given a subset of its expansion coefficients in a special non-orthogonal basis. This basis and its dual basis--the central elements of our model--are explicitly derived. Our analysis is grounded in a specific centering of the points that is unique to the Nystr\"om method. With this in mind, we extend previous work in Euclidean distance geometry by providing a general dual basis approach for points centered anywhere.

A dual basis approach to multidimensional scaling: spectral analysis and graph regularity

Classical multidimensional scaling (CMDS) is a technique that aims to embed a set of objects in a Euclidean space given their pairwise Euclidean distance matrix. The main part of CMDS is based on double centering a squared distance matrix and employing a truncated eigendecomposition to recover the point coordinates. A central result in CMDS connects the squared Euclidean matrix to a Gram matrix derived from the set of points. In this paper, we study a dual basis approach to classical multidimensional scaling. We give an explicit formula for the dual basis and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.

Sparse, Geometric Autoencoder Models of V1

The classical sparse coding model represents visual stimuli as a linear combination of a handful of learned basis functions that are Gabor-like when trained on natural image data. However, the Gabor-like filters learned by classical sparse coding far overpredict well-tuned simple cell receptive field (SCRF) profiles. A number of subsequent models have either discarded the sparse dictionary learning framework entirely or have yet to take advantage of the surge in unrolled, neural dictionary learning architectures. A key missing theme of these updates is a stronger notion of \emph{structured sparsity}. We propose an autoencoder architecture whose latent representations are implicitly, locally organized for spectral clustering, which begets artificial neurons better matched to observed primate data. The weighted-$\ell_1$ (WL) constraint in the autoencoder objective function maintains core ideas of the sparse coding framework, yet also offers a promising path to describe the differentiation of receptive fields in terms of a discriminative hierarchy in future work.

Alternating minimization algorithm with initialization analysis for r-local and k-sparse unlabeled sensing

The unlabeled sensing problem is to recover an unknown signal from permuted linear measurements. We propose an alternating minimization algorithm with a suitable initialization for the widely considered k-sparse permutation model. Assuming either a Gaussian measurement matrix or a sub-Gaussian signal, we upper bound the initialization error for the r-local and k-sparse permutation models in terms of the block size $r$ and the number of shuffles k, respectively. Our algorithm is computationally scalable and, compared to baseline methods, achieves superior performance on real and synthetic datasets.

A Recommender System for Equitable Public Art Curation and Installation

The placement of art in public spaces can have a significant impact on who feels a sense of belonging. In cities, public art communicates whose interests and culture are being favored. In this paper, we propose a graph matching approach with local constraints to build a curatorial tool for selecting public art in a way that supports inclusive spaces. We develop a cost matrix by drawing on Schelling's model of segregation. Using the cost matrix as an input, the optimization problem is solved via projected gradient descent to obtain a soft assignment matrix. We discuss regularization terms to set curatorial constraints. Our optimization program allocates artwork to public spaces and walls in a way that de-prioritizes "in-group" preferences, by satisfying minimum representation and exposure criteria. We draw on existing literature to develop a fairness metric for our algorithmic output. Using Tufts University as a testbed, we assess the effectiveness of our approach and discuss its potential pitfalls from both a curatorial and equity standpoint.

Measure Estimation in the Barycentric Coding Model

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycenteric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.

r-local sensing: Improved algorithm and applications

The unlabeled sensing problem is to solve a noisy linear system of equations under unknown permutation of the measurements. We study a particular case of the problem where the permutations are restricted to be r-local, i.e. the permutation matrix is block diagonal with r x r blocks. Assuming a Gaussian measurement matrix, we argue that the r-local permutation model is more challenging compared to a recent sparse permutation model. We propose a proximal alternating minimization algorithm for the general unlabeled sensing problem that provably converges to a first order stationary point. Applied to the r-local model, we show that the resulting algorithm is efficient. We validate the algorithm on synthetic and real datasets. We also formulate the 1-d unassigned distance geometry problem as an unlabeled sensing problem with a structured measurement matrix.

Weighed $\ell_1$ on the simplex: Compressive sensing meets locality

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard $\ell_1$ minimization problem. We propose weighted $\ell_0$ and weighted $\ell_1$ metrics that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted $\ell_1$ and weighted $\ell_0$. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.

On the convergence of group-sparse autoencoders

Recent approaches in the theoretical analysis of model-based deep learning architectures have studied the convergence of gradient descent in shallow ReLU networks that arise from generative models whose hidden layers are sparse. Motivated by the success of architectures that impose structured forms of sparsity, we introduce and study a group-sparse autoencoder that accounts for a variety of generative models, and utilizes a group-sparse ReLU activation function to force the non-zero units at a given layer to occur in blocks. For clustering models, inputs that result in the same group of active units belong to the same cluster. We proceed to analyze the gradient dynamics of a shallow instance of the proposed autoencoder, trained with data adhering to a group-sparse generative model. In this setting, we theoretically prove the convergence of the network parameters to a neighborhood of the generating matrix. We validate our model through numerical analysis and highlight the superior performance of networks with a group-sparse ReLU compared to networks that utilize traditional ReLUs, both in sparse coding and in parameter recovery tasks. We also provide real data experiments to corroborate the simulated results, and emphasize the clustering capabilities of structured sparsity models.