arXiv subjects

Abiy Tasissa

Publications and source records attributed to Abiy Tasissa.

At least 19 recordsLinked to original sources

arXiv2026

Provable Non-Convex Euclidean Distance Matrix Completion: Geometry, Reconstruction, and Robustness

The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network localization, molecular conformation, and manifold learning. In this paper, we propose a Riemannian optimization framework for solving the EDMC problem by formulating it as a low-rank matrix completion task over the space of positive semi-definite Gram matrices. The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through nonnegativity and the triangle inequality, a structure inherited from classical multidimensional scaling. Under a Bernoulli sampling model for observed distances, we prove that Riemannian gradient descent on the manifold of rank-$r$ matrices locally converges linearly with high probability when the sampling probability satisfies $p\geq O(ν^2 r^2\log(n)/n)$, where $ν$ is an EDMC-specific incoherence parameter. Furthermore, we provide an initialization candidate using a one-step hard thresholding procedure that yields convergence, provided the sampling probability satisfies $p \geq O(νr^{3/2}\log^{3/4}(n)/n^{1/4})$. A key technical contribution of this work is the analysis of a symmetric linear operator arising from a dual basis expansion in the non-orthogonal basis, which requires analysis of a second order degenerate U-statistic to establish an optimal restricted isometry property in the presence of coupled terms. Empirical evaluations on synthetic data demonstrate that our algorithm achieves competitive performance relative to state-of-the-art methods. Moreover, we provide a geometric interpretation of matrix incoherence tailored to the EDMC setting and provide robustness guarantees for our method.

math.OC↗

arXiv2026

Discriminative reconstruction via simultaneous dense and sparse coding

Discriminative features extracted from the sparse coding model have been shown to perform well for classification. Recent deep learning architectures have further improved reconstruction in inverse problems by considering new dense priors learned from data. We propose a novel dense and sparse coding model that integrates both representation capability and discriminative features. The model studies the problem of recovering a dense vector $\mathbf{x}$ and a sparse vector $\mathbf{u}$ given measurements of the form $\mathbf{y} = \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$. Our first analysis relies on a geometric condition, specifically the minimal angle between the spanning subspaces of matrices $\mathbf{A}$ and $\mathbf{B}$, which ensures a unique solution to the model. The second analysis shows that, under some conditions on $\mathbf{A}$ and $\mathbf{B}$, a convex program recovers the dense and sparse components. We validate the effectiveness of the model on simulated data and propose a dense and sparse autoencoder (DenSaE) tailored to learning the dictionaries from the dense and sparse model. We demonstrate that (i) DenSaE denoises natural images better than architectures derived from the sparse coding model ($\mathbf{B}\mathbf{u}$), (ii) in the presence of noise, training the biases in the latter amounts to implicitly learning the $\mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$ model, (iii) $\mathbf{A}$ and $\mathbf{B}$ capture low- and high-frequency contents, respectively, and (iv) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation.

cs.IT↗

arXiv2026

Shift-Invariant Feature Attribution in the Application of Wireless Electrocardiograms

Assigning relevance scores to the input features of a machine learning model enables to measure the contributions of the features in achieving a correct outcome. It is regarded as one of the approaches towards developing explainable models. For biomedical assignments, this is very useful for medical experts to comprehend machine-based decisions. In the analysis of electro cardiogram (ECG) signals, in particular, understanding which of the electrocardiogram samples or features contributed most for a given decision amounts to understanding the underlying cardiac phases or conditions the machine tries to explain. For the computation of relevance scores, determining the proper baseline is important. Moreover, the scores should have a distribution which is at once intuitive to interpret and easy to associate with the underline cardiac reality. The purpose of this work is to achieve these goals. Specifically, we propose a shift-invariant baseline which has a physical significance in the analysis as well as interpretation of electrocardiogram measurements. Moreover, we aggregate significance scores in such a way that they can be mapped to cardiac phases. We demonstrate our approach by inferring physical exertion from cardiac exertion using a residual network. We show that the ECG samples which achieved the highest relevance scores (and, therefore, which contributed most to the accurate recognition of the physical exertion) are those associated with the P and T waves. Index Terms Attribution, baseline, cardiovascular diseases, electrocardiogram, activity recognition, machine learning

eess.SP↗

arXiv2025

Recovering Wasserstein Distance Matrices from Few Measurements

This paper proposes two algorithms for estimating square Wasserstein distance matrices from a small number of entries. These matrices are used to compute manifold learning embeddings like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are extremely costly to compute. We analyze matrix completion from upper triangular samples and Nyström completion in which $\mathcal{O}(d\log(d))$ columns of the distance matrices are computed where $d$ is the desired embedding dimension, prove stability of MDS under Nyström completion, and show that it can outperform matrix completion for a fixed budget of sample distances. Finally, we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is stable on data embedded from the Nyström estimation of the distance matrix even when only 10\% of the columns are computed.

stat.ML↗

arXiv2025

Robust Node Localization for Rough and Extreme Deployment Environments

Many applications have been identified which require the deployment of large-scale low-power wireless sensor networks. Some of the deployment environments, however, impose harsh operation conditions due to intense cross-technology interference, extreme weather conditions (heavy rainfall, excessive heat, etc.), or rough motion, thereby affecting the quality and predictability of the wireless links the nodes establish. In localization tasks, these conditions often lead to significant errors in estimating the position of target nodes. Motivated by the practical deployments of sensors on the surface of different water bodies, we address the problem of identifying susceptible nodes and robustly estimating their positions. We formulate these tasks as a compressive sensing problem and propose algorithms for both node identification and robust estimation. Additionally, we design an optimal anchor configuration to maximize the robustness of the position estimation task. Our numerical results and comparisons with competitive methods demonstrate that the proposed algorithms achieve both objectives with a modest number of anchors. Since our method relies only on target-to-anchor distances, it is broadly applicable and yields resilient, robust localization.

eess.SP↗

arXiv2025

A Dual Basis Approach for Structured Robust Euclidean Distance Geometry

Euclidean Distance Matrix (EDM), which consists of pairwise squared Euclidean distances of a given point configuration, finds many applications in modern machine learning. This paper considers the setting where only a set of anchor nodes is used to collect the distances between themselves and the rest. In the presence of potential outliers, it results in a structured partial observation on EDM with partial corruptions. Note that an EDM can be connected to a positive semi-definite Gram matrix via a non-orthogonal dual basis. Inspired by recent development of non-orthogonal dual basis in optimization, we propose a novel algorithmic framework, dubbed Robust Euclidean Distance Geometry via Dual Basis (RoDEoDB), for recovering the Euclidean distance geometry, i.e., the underlying point configuration. The exact recovery guarantees have been established in terms of both the Gram matrix and point configuration, under some mild conditions. Empirical experiments show superior performance of RoDEoDB on sensor localization and molecular conformation datasets.

cs.LG↗

arXiv2025

Prediction of the Received Power of Low-Power Networks Using Inertial Sensors

Low-power and cost-effective IoT sensing nodes enable scalable monitoring of different environments. Some of these environments impose rough and extreme operating conditions, requiring continuous adaptation and reconfiguration of physical and link layer parameters. In this paper, we closely investigate the stability of the wireless links established between nodes deployed on the surface of different water bodies and propose a model to predict the received power. Our model is based on Minimum Mean Square Estimation (MMSE) and relies on the statistics of received power and the motion the nodes experience during communication. One of the drawbacks of MMSE is its reliance on matrix inversion, which is at once computationally expensive and difficult to implement with resource constrained devices. We forgo this stage by estimating model parameters using the gradient-descent approach, which is much simpler to implement. The model achieves a prediction accuracy of 91% even with a small number of iterations.

cs.NI↗

arXiv2025

Structured Sampling for Robust Euclidean Distance Geometry

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 Nyström 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.

cs.LG↗

arXiv2024

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

Unlabeled sensing is a linear inverse problem with permuted measurements. We propose an alternating minimization (AltMin) algorithm with a suitable initialization for two widely considered permutation models: partially shuffled/$k$-sparse permutations and $r$-local/block diagonal permutations. Key to the performance of the AltMin algorithm is the initialization. For the exact unlabeled sensing problem, assuming either a Gaussian measurement matrix or a sub-Gaussian signal, we bound the initialization error in terms of the number of blocks $s$ and the number of shuffles $k$. Experimental results show that our algorithm is fast, applicable to both permutation models, and robust to choice of measurement matrix. We also test our algorithm on several real datasets for the linked linear regression problem and show superior performance compared to baseline methods.

eess.SP↗

arXiv2024

Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization

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.

cs.LG↗

arXiv2024

Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees

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.

math.OC↗

arXiv2024

Weighed l1 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 $\ell_0$ and weighted $\ell_1$. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.

eess.SP↗

arXiv2024

K-Deep Simplex: Deep Manifold Learning via Local Dictionaries

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 employs a 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 unrolling. 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. We further show the stability of the representation coefficients under mild geometrical assumptions. If the representation coefficients are fixed, we prove that the sub-problem of minimizing over the dictionary yields a unique solution. Further, we show that low-dimensional representations can be efficiently obtained from the covariance of the coefficient matrix. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.

cs.LG↗

arXiv2024

A dual basis approach to multidimensional scaling

Classical multidimensional scaling (CMDS) is a technique that embeds a set of objects in a Euclidean space given their pairwise Euclidean distances. The main part of CMDS involves double centering a squared distance matrix and using a truncated eigendecomposition to recover the point coordinates. In this paper, motivated by a study in Euclidean distance geometry, we explore a dual basis approach to CMDS. We give an explicit formula for the dual basis vectors and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.

math.SP↗

arXiv2024

Localization from structured distance matrices via low-rank matrix recovery

We study the problem of determining the configuration of $n$ points by using their distances to $m$ nodes, referred to as anchor nodes. One sampling scheme is Nystrom sampling, which assumes known distances between the anchors and between the anchors and the $n$ points, while the distances among the $n$ points are unknown. For this scheme, a simple adaptation of the Nystrom method, which is often used for kernel approximation, is a viable technique to estimate the configuration of the anchors and the $n$ points. In this manuscript, we propose a modified version of Nystrom sampling, where the distances from every node to one central node are known, but all other distances are incomplete. In this setting, the standard Nystrom approach is not applicable, necessitating an alternative technique to estimate the configuration of the anchors and the $n$ points. We show that this problem can be framed as the recovery of a low-rank submatrix of a Gram matrix. Using synthetic and real data, we demonstrate that the proposed approach can exactly recover configurations of points given sufficient distance samples. This underscores that, in contrast to methods that rely on global sampling of distance matrices, the task of estimating the configuration of points can be done efficiently via structured sampling with well-chosen reliable anchors. Finally, our main analysis is grounded in a specific centering of the points. With this in mind, we extend previous work in Euclidean distance geometry by providing a general dual basis approach for points centered anywhere.

cs.IT↗

arXiv2024

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.

cs.LG↗

arXiv2024

RACH-Space: Reconstructing Adaptive Convex Hull Space with Applications in Weak Supervision

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.

cs.LG↗

arXiv2023

Clustering Inductive Biases with Unrolled Networks

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.

q-bio.NC↗