arXiv subjects

Bubacarr Bah

Publications and source records attributed to Bubacarr Bah.

At least 19 recordsLinked to original sources

arXiv2024

Physics-informed neural networks for Timoshenko system with Thermoelasticity

The main focus of this paper is to analyze the behavior of a numerical solution of the Timoshenko system coupled with Thermoelasticity and incorporating second sound effects. In order to address this target, we employ the Physics-Informed Neural Networks (PINNs) framework to derive an approximate solution for the system. Our investigation delves into the extent to which this approximate solution can accurately capture the asymptotic behavior of the discrete energy, contingent upon the stability number $\chi$. Interestingly, the PINNs overcome the major difficulties encountered while using the standard numerical methods.

math.NA↗

arXiv2024

This actually looks like that: Proto-BagNets for local and global interpretability-by-design

Interpretability is a key requirement for the use of machine learning models in high-stakes applications, including medical diagnosis. Explaining black-box models mostly relies on post-hoc methods that do not faithfully reflect the model's behavior. As a remedy, prototype-based networks have been proposed, but their interpretability is limited as they have been shown to provide coarse, unreliable, and imprecise explanations. In this work, we introduce Proto-BagNets, an interpretable-by-design prototype-based model that combines the advantages of bag-of-local feature models and prototype learning to provide meaningful, coherent, and relevant prototypical parts needed for accurate and interpretable image classification tasks. We evaluated the Proto-BagNet for drusen detection on publicly available retinal OCT data. The Proto-BagNet performed comparably to the state-of-the-art interpretable and non-interpretable models while providing faithful, accurate, and clinically meaningful local and global explanations. The code is available at https://github.com/kdjoumessi/Proto-BagNets.

cs.AI↗

arXiv2023

Improved identification accuracy in equation learning via comprehensive $\boldsymbol{R^2}$-elimination and Bayesian model selection

In the field of equation learning, exhaustively considering all possible equations derived from a basis function dictionary is infeasible. Sparse regression and greedy algorithms have emerged as popular approaches to tackle this challenge. However, the presence of multicollinearity poses difficulties for sparse regression techniques, and greedy steps may inadvertently exclude terms of the true equation, leading to reduced identification accuracy. In this article, we present an approach that strikes a balance between comprehensiveness and efficiency in equation learning. Inspired by stepwise regression, our approach combines the coefficient of determination, $R^2$, and the Bayesian model evidence, $p(\boldsymbol y|\mathcal M)$, in a novel way. Our procedure is characterized by a comprehensive search with just a minor reduction of the model space at each iteration step. With two flavors of our approach and the adoption of $p(\boldsymbol y|\mathcal M)$ for bi-directional stepwise regression, we present a total of three new avenues for equation learning. Through three extensive numerical experiments involving random polynomials and dynamical systems, we compare our approach against four state-of-the-art methods and two standard approaches. The results demonstrate that our comprehensive search approach surpasses all other methods in terms of identification accuracy. In particular, the second flavor of our approach establishes an efficient overfitting penalty solely based on $R^2$, which achieves highest rates of exact equation recovery.

stat.ML↗

arXiv2023

Compressed sensing-based SARS-CoV-2 pool testing

We propose a compressed sensing-based testing approach with a practical measurement design and a tuning-free and noise-robust algorithm for detecting infected persons. Compressed sensing results can be used to provably detect a small number of infected persons among a possibly large number of people. There are several advantages of this method compared to classical group testing. Firstly, it is non-adaptive and thus possibly faster to perform than adaptive methods which is crucial in exponentially growing pandemic phases. Secondly, due to nonnegativity of measurements and an appropriate noise model, the compressed sensing problem can be solved with the non-negative least absolute deviation regression (NNLAD) algorithm. This convex tuning-free program requires the same number of tests as current state of the art group testing methods. Empirically it performs significantly better than theoretically guaranteed, and thus the high-throughput, reducing the number of tests to a fraction compared to other methods. Further, numerical evidence suggests that our method can correct sparsely occurring errors.

cs.IT↗

arXiv2023

A physics-informed neural network framework for modeling obstacle-related equations

Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g., TensorFlow or PyTorch. Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here extend PINNs to solve obstacle-related PDEs which present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of the solution that lies above a given obstacle. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.

cs.LG↗

arXiv2021

Efficient and Robust Mixed-Integer Optimization Methods for Training Binarized Deep Neural Networks

Compared to classical deep neural networks its binarized versions can be useful for applications on resource-limited devices due to their reduction in memory consumption and computational demands. In this work we study deep neural networks with binary activation functions and continuous or integer weights (BDNN). We show that the BDNN can be reformulated as a mixed-integer linear program with bounded weight space which can be solved to global optimality by classical mixed-integer programming solvers. Additionally, a local search heuristic is presented to calculate locally optimal networks. Furthermore to improve efficiency we present an iterative data-splitting heuristic which iteratively splits the training set into smaller subsets by using the k-mean method. Afterwards all data points in a given subset are forced to follow the same activation pattern, which leads to a much smaller number of integer variables in the mixed-integer programming formulation and therefore to computational improvements. Finally for the first time a robust model is presented which enforces robustness of the BDNN during training. All methods are tested on random and real datasets and our results indicate that all models can often compete with or even outperform classical DNNs on small network architectures confirming the viability for applications having restricted memory or computing power.

math.OC↗

arXiv2021

Towards the Localisation of Lesions in Diabetic Retinopathy

Convolutional Neural Networks (CNNs) have successfully been used to classify diabetic retinopathy (DR) fundus images in recent times. However, deeper representations in CNNs may capture higher-level semantics at the expense of spatial resolution. To make predictions usable for ophthalmologists, we use a post-attention technique called Gradient-weighted Class Activation Mapping (Grad-CAM) on the penultimate layer of deep learning models to produce coarse localisation maps on DR fundus images. This is to help identify discriminative regions in the images, consequently providing evidence for ophthalmologists to make a diagnosis and potentially save lives by early diagnosis. Specifically, this study uses pre-trained weights from four state-of-the-art deep learning models to produce and compare localisation maps of DR fundus images. The models used include VGG16, ResNet50, InceptionV3, and InceptionResNetV2. We find that InceptionV3 achieves the best performance with a test classification accuracy of 96.07%, and localise lesions better and faster than the other models.

cs.CV↗

arXiv2020

Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers

We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover, we establish that, for almost all initializations, the flow converges to a global minimum on the manifold of rank $k$ matrices for some $k\leq r$.

math.OC↗

arXiv2020

Efficient Tuning-Free $\ell_1$-Regression of Nonnegative Compressible Signals

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements. The general assumption is that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the $\ell_2$-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, nonnegative least squares recovers non-negative signals if the measurement matrix fulfills an additional property (sometimes called the $M^+$-criterion). However, if the measurement matrix is the biadjacency matrix of a random left regular bipartite graph it obeys with a high probability the null space property with respect to the $\ell_1$-norm with optimal parameters. Therefore, we discuss non-negative least absolute deviation (NNLAD). For these measurement matrices, we prove a uniform, stable and robust recovery guarantee without the need for tuning. Such guarantees are important, since binary expander matrices are sparse and thus allow for fast sketching and recovery. We will further present a method to solve the NNLAD numerically and show that this is comparable to state of the art methods. Lastly, we explain how the NNLAD can be used for viral detection in the recent COVID-19 crisis.

cs.IT↗

arXiv2020

An Integer Programming Approach to Deep Neural Networks with Binary Activation Functions

We study deep neural networks with binary activation functions (BDNN), i.e. the activation function only has two states. We show that the BDNN can be reformulated as a mixed-integer linear program which can be solved to global optimality by classical integer programming solvers. Additionally, a heuristic solution algorithm is presented and we study the model under data uncertainty, applying a two-stage robust optimization approach. We implemented our methods on random and real datasets and show that the heuristic version of the BDNN outperforms classical deep neural networks on the Breast Cancer Wisconsin dataset while performing worse on random data.

math.OC↗

arXiv2020

On Error Correction Neural Networks for Economic Forecasting

Recurrent neural networks (RNNs) are more suitable for learning non-linear dependencies in dynamical systems from observed time series data. In practice all the external variables driving such systems are not known a priori, especially in economical forecasting. A class of RNNs called Error Correction Neural Networks (ECNNs) was designed to compensate for missing input variables. It does this by feeding back in the current step the error made in the previous step. The ECNN is implemented in Python by the computation of the appropriate gradients and it is tested on stock market predictions. As expected it out performed the simple RNN and LSTM and other hybrid models which involve a de-noising pre-processing step. The intuition for the latter is that de-noising may lead to loss of information.

cs.LG↗

arXiv2020

Discrete Optimization Methods for Group Model Selection in Compressed Sensing

In this article we study the problem of signal recovery for group models. More precisely for a given set of groups, each containing a small subset of indices, and for given linear sketches of the true signal vector which is known to be group-sparse in the sense that its support is contained in the union of a small number of these groups, we study algorithms which successfully recover the true signal just by the knowledge of its linear sketches. We derive model projection complexity results and algorithms for more general group models than the state-of-the-art. We consider two versions of the classical Iterative Hard Thresholding algorithm (IHT). The classical version iteratively calculates the exact projection of a vector onto the group model, while the approximate version (AM-IHT) uses a head- and a tail-approximation iteratively. We apply both variants to group models and analyse the two cases where the sensing matrix is a Gaussian matrix and a model expander matrix. To solve the exact projection problem on the group model, which is known to be equivalent to the maximum weight coverage problem, we use discrete optimization methods based on dynamic programming and Benders' Decomposition. The head- and tail-approximations are derived by a classical greedy-method and LP-rounding, respectively.

math.OC↗

arXiv2018

On the construction of sparse matrices from expander graphs

We revisit the asymptotic analysis of probabilistic construction of adjacency matrices of expander graphs proposed in [4]. With better bounds we derived a new reduced sample complexity for the number of nonzeros per column of these matrices, precisely $d = \mathcal{O}\left(\log_s(N/s) \right)$; as opposed to the standard $d = \mathcal{O}\left(\log(N/s) \right)$. This gives insights into why using small $d$ performed well in numerical experiments involving such matrices. Furthermore, we derive quantitative sampling theorems for our constructions which show our construction outperforming the existing state-of-the-art. We also used our results to compare performance of sparse recovery algorithms where these matrices are used for linear sketching.

cs.IT↗

arXiv2016

Sparse matrices for weighted sparse recovery

We derived the first sparse recovery guarantees for weighted $\ell_1$ minimization with sparse random matrices and the class of weighted sparse signals, using a weighted versions of the null space property to derive these guarantees. These sparse matrices from expender graphs can be applied very fast and have other better computational complexities than their dense counterparts. In addition we show that, using such sparse matrices, weighted sparse recovery with weighted $\ell_1$ minimization leads to sample complexities that are linear in the weighted sparsity of the signal and these sampling rates can be smaller than those of standard sparse recovery. Moreover, these results reduce to known results in standard sparse recovery and sparse recovery with prior information and the results are supported by numerical experiments.

math.NA↗

arXiv2016

Convex block-sparse linear regression with expanders -- provably

Sparse matrices are favorable objects in machine learning and optimization. When such matrices are used, in place of dense ones, the overall complexity requirements in optimization can be significantly reduced in practice, both in terms of space and run-time. Prompted by this observation, we study a convex optimization scheme for block-sparse recovery from linear measurements. To obtain linear sketches, we use expander matrices, i.e., sparse matrices containing only few non-zeros per column. Hitherto, to the best of our knowledge, such algorithmic solutions have been only studied from a non-convex perspective. Our aim here is to theoretically characterize the performance of convex approaches under such setting. Our key novelty is the expression of the recovery error in terms of the model-based norm, while assuring that solution lives in the model. To achieve this, we show that sparse model-based matrices satisfy a group version of the null-space property. Our experimental findings on synthetic and real applications support our claims for faster recovery in the convex setting -- as opposed to using dense sensing matrices, while showing a competitive recovery performance.

cs.IT↗

arXiv2015

The sample complexity of weighted sparse approximation

For Gaussian sampling matrices, we provide bounds on the minimal number of measurements $m$ required to achieve robust weighted sparse recovery guarantees in terms of how well a given prior model for the sparsity support aligns with the true underlying support. Our main contribution is that for a sparse vector ${\bf x} \in \mathbb{R}^N$ supported on an unknown set $\mathcal{S} \subset \{1, \dots, N\}$ with $|\mathcal{S}|\leq k$, if $\mathcal{S}$ has \emph{weighted cardinality} $\omega(\mathcal{S}) := \sum_{j \in \mathcal{S}} \omega_j^2$, and if the weights on $\mathcal{S}^c$ exhibit mild growth, $\omega_j^2 \geq \gamma \log(j/\omega(\mathcal{S}))$ for $j\in\mathcal{S}^c$ and $\gamma > 0$, then the sample complexity for sparse recovery via weighted $\ell_1$-minimization using weights $\omega_j$ is linear in the weighted sparsity level, and $m = \mathcal{O}(\omega(\mathcal{S})/\gamma)$. This main result is a generalization of special cases including a) the standard sparse recovery setting where all weights $\omega_j \equiv 1$, and $m = \mathcal{O}\left(k\log\left(N/k\right)\right)$; b) the setting where the support is known a priori, and $m = \mathcal{O}(k)$; and c) the setting of sparse recovery with prior information, and $m$ depends on how well the weights are aligned with the support set $\mathcal{S}$. We further extend the results in case c) to the setting of additive noise. Our results are {\em nonuniform} that is they apply for a fixed support, unknown a priori, and the weights on $\mathcal{S}$ do not all have to be smaller than the weights on $\mathcal{S}^c$ for our recovery results to hold.

math.NA↗

arXiv2014

Model-based Sketching and Recovery with Expanders

Linear sketching and recovery of sparse vectors with randomly constructed sparse matrices has numerous applications in several areas, including compressive sensing, data stream computing, graph sketching, and combinatorial group testing. This paper considers the same problem with the added twist that the sparse coefficients of the unknown vector exhibit further correlations as determined by a known sparsity model. We prove that exploiting model-based sparsity in recovery provably reduces the sketch size without sacrificing recovery quality. In this context, we present the model-expander iterative hard thresholding algorithm for recovering model sparse signals from linear sketches obtained via sparse adjacency matrices of expander graphs with rigorous performance guarantees. The main computational cost of our algorithm depends on the difficulty of projecting onto the model-sparse set. For the tree and group-based sparsity models we describe in this paper, such projections can be obtained in linear time. Finally, we provide numerical experiments to illustrate the theoretical results in action.

math.NA↗

arXiv2013

On construction and analysis of sparse random matrices and expander graphs with applications to compressed sensing

We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random. These matrices have a one-to-one correspondence with the adjacency matrices of lossless expander graphs. We present tail bounds on the probability that the cardinality of the set of neighbors for these graphs will be less than the expected value. The bounds are derived through the analysis of collisions in unions of sets using a {\em dyadic splitting} technique. This analysis led to the derivation of better constants that allow for quantitative theorems on existence of lossless expander graphs and hence the sparse random matrices we consider and also quantitative compressed sensing sampling theorems when using sparse non mean-zero measurement matrices.

cs.IT↗