This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nystr\"om method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples. To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided. As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art.

The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms grounded in the Riemannian optimization framework to address the EDG problem. Our approach formulates the problem as a low-rank matrix completion task over the Gram matrix, using partial measurements represented as expansion coefficients of the Gram matrix in a non-orthogonal basis. For the first algorithm, under a uniform sampling with replacement model for the observed distance entries, we demonstrate that, with high probability, a Riemannian gradient-like algorithm on the manifold of rank-$r$ matrices converges linearly to the true solution, given initialization via a one-step hard thresholding. This holds provided the number of samples, $m$, satisfies $m \geq \mathcal{O}(n^{7/4}r^2 \log(n))$. With a more refined initialization, achieved through resampled Riemannian gradient-like descent, we further improve this bound to $m \geq \mathcal{O}(nr^2 \log(n))$. Our analysis for the first algorithm leverages a non-self-adjoint operator and depends on deriving eigenvalue bounds for an inner product matrix of restricted basis matrices, leveraging sparsity properties for tighter guarantees than previously established. The second algorithm introduces a self-adjoint surrogate for the sampling operator. This algorithm demonstrates strong numerical performance on both synthetic and real data. Furthermore, we show that optimizing over manifolds of higher-than-rank-$r$ matrices yields superior numerical results, consistent with recent literature on overparameterization in the EDG problem.

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.

We introduce RACH-Space, an algorithm for labelling unlabelled data in weakly supervised learning, given incomplete, noisy information about the labels. RACH-Space offers simplicity in implementation without requiring hard assumptions on data or the sources of weak supervision, and is well suited for practical applications where fully labelled data is not available. Our method is built upon a geometrical interpretation of the space spanned by the set of weak signals. We also analyze the theoretical properties underlying the relationship between the convex hulls in this space and the accuracy of our output labels, bridging geometry with machine learning. Empirical results demonstrate that RACH-Space works well in practice and compares favorably to the best existing label models for weakly supervised learning.

The classical sparse coding (SC) 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 profiles observed empirically. While neurons fire sparsely, neuronal populations are also organized in physical space by their sensitivity to certain features. In V1, this organization is a smooth progression of orientations along the cortical sheet. A number of subsequent models have either discarded the sparse dictionary learning framework entirely or whose updates 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 (WLSC) whose latent representations are implicitly, locally organized for spectral clustering through a Laplacian quadratic form of a bipartite graph, which generates a diverse set of artificial receptive fields that match primate data in V1 as faithfully as recent contrastive frameworks like Local Low Dimensionality, or LLD \citep{lld} that discard sparse dictionary learning. By unifying sparse and smooth coding in models of the early visual cortex through our autoencoder, we also show that our regularization can be interpreted as early-stage specialization of receptive fields to certain classes of stimuli; that is, we induce a weak clustering bias for later stages of cortex where functional and spatial segregation (i.e. topography) are known to occur. The results show an imperative for \emph{spatial regularization} of both the receptive fields and firing rates to begin to describe feature disentanglement in V1 and beyond.

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.

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.

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.

We propose K-Deep Simplex (KDS) which, given a set of data points, learns a dictionary comprising synthetic landmarks, along with representation coefficients supported on a simplex. KDS integrates manifold learning and sparse coding/dictionary learning: reconstruction term, as in classical dictionary learning, and a novel local weighted $\ell_1$ penalty that encourages each data point to represent itself as a convex combination of nearby landmarks. We solve the proposed optimization program using alternating minimization and design an efficient, interpretable autoencoder using algorithm enrolling. We theoretically analyze the proposed program by relating the weighted $\ell_1$ penalty in KDS to a weighted $\ell_0$ program. Assuming that the data are generated from a Delaunay triangulation, we prove the equivalence of the weighted $\ell_1$ and weighted $\ell_0$ programs. If the representation coefficients are given, we prove that the resulting dictionary is unique. Further, we show that low-dimensional representations can be efficiently obtained from the covariance of the coefficient matrix. We apply KDS to the unsupervised clustering problem and prove theoretical performance guarantees. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.