arXiv subjects

Issa Karambal

Publications and source records attributed to Issa Karambal.

12 recordsLinked to original sources

arXiv2026

Stochastic Lanczos Quadrature for Computational Uncertainty in Linear Algebra

Applying a matrix function to a vector is a common operation in large-scale scientific computing, such as Bayesian inference. When the matrix is symmetric positive definite and the function is its inverse, the Lanczos algorithm gives a matrix-free approximation from matrix-vector products alone. Stopping after a fixed number of steps keeps the approximation on the explored Krylov subspace but assigns nothing to its orthogonal complement, which holds the smallest eigenvalues and therefore the largest contributions to the inverse. This truncation is usually treated as a silent numerical error rather than as measurable uncertainty. We show that a single Lanczos run yields both a matrix-free approximation of the inverse and a Gaussian distribution over the exact result, whose covariance measures what the truncation discarded and concentrates as the number of steps grows. The covariance restores the complement in two parts: a rank-one coupling at the truncation boundary, given exactly by the Lanczos residual, and an isotropic term on the remaining bulk, whose size we estimate by Projected Stochastic Lanczos Quadrature, reusing one Krylov basis across all probes. We prove consistency, with bias decaying exponentially in the quadrature depth and variance at the Monte Carlo rate

math.NA↗

arXiv2026

ReliableNet: A Chance-Constrained Approach to Trustworthy Classification in Deep Learning

A prediction that is both confident and wrong is a critical reliability failure because it can bypass abstention and human review precisely when the model is mistaken. Empirical risk minimization (ERM) controls average loss but not this failure directly, while calibration, uncertainty estimation, conformal risk control, and selective prediction methods target related reliability properties rather than bounding the joint failure event during training. We propose ReliableNet, which constrains the Joint Confident-Wrong (JCW) probability, the probability that a prediction is simultaneously confident and incorrect, below a user-specified risk budget $\alpha\in(0,1)$. We formulate this as a chance-constrained ERM problem, use a conservative smooth inner approximation whose population feasibility implies the original JCW constraint. Across four tabular and two image datasets, ReliableNet is the only method certified within the JCW budget for every dataset and seed in distribution, when compared against baselines spanning ERM, post-hoc calibration, conformal risk control, and selective prediction. Under demographic, ambiguity, spurious-correlation, novel-class, and covariate shifts, it achieves the lowest empirical JCW among the compared methods while remaining very competitive in accuracy, coverage, calibration, and selective prediction. Risk-coverage results further indicate that ReliableNet achieves better selective ranking than the benchmark methods on most datasets. Overall, ReliableNet provides a principled approach to trustworthy classification.

cs.LG↗

arXiv2026

Optimal Stochastic Krylov based Techniques for Large- Scale Log-Determinant Estimation

Estimating the logarithm of the determinant of large sparse positive definite symmetric matrices is an important task in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. In this work, we introduce two scalable and efficient methods for large-scale log-determinant termed the Optimal Stochastic Arnoldi with Incomplete Orthogonalization Procedure (OSA-IOP) and the Optimal Stochastic Lanczos Quadrature (OSLQ). The OSA-IOP approach extends the Incomplete Orthogonalization Procedure (IOP), originally developed for matrix exponential functions for exponential time stepping integrators, to compute the action of the matrix algorithm on a vector. We observe that combining IOP with a randomized Hutch++ algorithm, the OSA-IOP significantly reduces computational cost while maintaining high accuracy. The OSLQ method estimates log-determinants by coupling Lanczos quadrature with Hutch++ and controlled orthogonalization, leveraging Krylov subspaces as efficient quadrature mechanisms to approximate quadratic forms involving the matrix logarithm. We derive error bounds for both methods. Extensive numerical experiments on large-scale sparse matrices from real-world applications demonstrate the accuracy, robustness, and scalability of the proposed approaches.

math.NA↗

arXiv2026

Novel technique based on L\'eja Points Approximation for Log-determinant Estimation of Large matrices

The computation of the Log-determinant of large, sparse, symmetric positive definite (SPD) matrices is essential in many scientific computational fields such as numerical linear algebra and machine learning. In low dimensions, Cholesky is preferred, but in high dimensions, its computation may be prohibitive due to memory limitation. To circumvent this, Krylov subspace techniques have proven to be efficient but may be computationally expensive due to the required orthogonalization processes. In this paper, we introduce a novel technique to estimate the Log-determinant of a matrix using L\'eja points, where the implementation is only based on matrix multiplications and a rough estimation of eigenvalue bounds of the matrix. By coupling L\'eja points interpolation with a randomized algorithm called Hutch++, we achieve substantial reductions in computational complexity while preserving significant accuracy compared to the stochastic Lanczos quadrature. We establish the approximation errors of the matrix function together with multiplicative error bounds for the approximations obtained by this method. The effectiveness and scalability of the proposed method on both large sparse synthetic matrices (maximum likelihood in Gaussian Markov Random fields) and large-scale real-world matrices are confirmed through numerical experiments.

math.NA↗

arXiv2025

Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equations

Recent studies have demonstrated the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). Source inversion problems under sparse measurements for parabolic partial differential equations (PDEs) are particularly challenging to solve using PINNs, due to their severe ill-posedness and the multiple unknowns involved including the source function and the PDE parameters. Although the neural tangent kernel (NTK) of PINNs has been widely used in forward problems involving a single neural network, its extension to inverse problems involving multiple neural networks remains less explored. In this work, we propose a weighted adaptive approach based on the NTK of PINNS including multiple separate networks representing the solution, the unknown source, and the PDE parameters. The key idea behind our methodology is to simultaneously solve the joint recovery of the solution, the source function along with the unknown parameters thereby using the underlying partial differential equation as a constraint that couples multiple unknown functional parameters, leading to more efficient use of the limited information in the measurements. We apply our method on the advection-diffusion equation and we present various 2D and 3D numerical experiments using different types of measurements data that reflect practical engineering systems. Our proposed method is successful in estimating the unknown source function, the velocity and diffusion parameters as well as recovering the solution of the equation, while remaining robust to additional noise in the measurements.

stat.ML↗

arXiv2025

RRaPINNs: Residual Risk-Aware Physics Informed Neural Networks

Physics-informed neural networks (PINNs) typically minimize average residuals, which can conceal large, localized errors. We propose Residual Risk-Aware Physics-Informed Neural Networks PINNs (RRaPINNs), a single-network framework that optimizes tail-focused objectives using Conditional Value-at-Risk (CVaR), we also introduced a Mean-Excess (ME) surrogate penalty to directly control worst-case PDE residuals. This casts PINN training as risk-sensitive optimization and links it to chance-constrained formulations. The method is effective and simple to implement. Across several partial differential equations (PDEs) such as Burgers, Heat, Korteweg-de-Vries, and Poisson (including a Poisson interface problem with a source jump at x=0.5) equations, RRaPINNs reduce tail residuals while maintaining or improving mean errors compared to vanilla PINNs, Residual-Based Attention and its variant using convolution weighting; the ME surrogate yields smoother optimization than a direct CVaR hinge. The chance constraint reliability level $\alpha$ acts as a transparent knob trading bulk accuracy (lower $\alpha$ ) for stricter tail control (higher $\alpha$ ). We discuss the framework limitations, including memoryless sampling, global-only tail budgeting, and residual-centric risk, and outline remedies via persistent hard-point replay, local risk budgets, and multi-objective risk over BC/IC terms. RRaPINNs offer a practical path to reliability-aware scientific ML for both smooth and discontinuous PDEs.

cs.LG↗

arXiv2025

The Reflexive Integrated Information Unit: A Differentiable Primitive for Artificial Consciousness

Research on artificial consciousness lacks the equivalent of the perceptron: a small, trainable module that can be copied, benchmarked, and iteratively improved. We introduce the Reflexive Integrated Information Unit (RIIU), a recurrent cell that augments its hidden state $h$ with two additional vectors: (i) a meta-state $\mu$ that records the cell's own causal footprint, and (ii) a broadcast buffer $B$ that exposes that footprint to the rest of the network. A sliding-window covariance and a differentiable Auto-$\Phi$ surrogate let each RIIU maximize local information integration online. We prove that RIIUs (1) are end-to-end differentiable, (2) compose additively, and (3) perform $\Phi$-monotone plasticity under gradient ascent. In an eight-way Grid-world, a four-layer RIIU agent restores $>90\%$ reward within 13 steps after actuator failure, twice as fast as a parameter-matched GRU, while maintaining a non-zero Auto-$\Phi$ signal. By shrinking "consciousness-like" computation down to unit scale, RIIUs turn a philosophical debate into an empirical mathematical problem.

cs.AI↗

arXiv2025

Localized Weather Prediction Using Kolmogorov-Arnold Network-Based Models and Deep RNNs

Weather forecasting is crucial for managing risks and economic planning, particularly in tropical Africa, where extreme events severely impact livelihoods. Yet, existing forecasting methods often struggle with the region's complex, non-linear weather patterns. This study benchmarks deep recurrent neural networks such as $\texttt{LSTM, GRU, BiLSTM, BiGRU}$, and Kolmogorov-Arnold-based models $(\texttt{KAN} and \texttt{TKAN})$ for daily forecasting of temperature, precipitation, and pressure in two tropical cities: Abidjan, Cote d'Ivoire (Ivory Coast) and Kigali (Rwanda). We further introduce two customized variants of $ \texttt{TKAN}$ that replace its original $\texttt{SiLU}$ activation function with $ \texttt{GeLU}$ and \texttt{MiSH}, respectively. Using station-level meteorological data spanning from 2010 to 2024, we evaluate all the models on standard regression metrics. $\texttt{KAN}$ achieves temperature prediction ($R^2=0.9986$ in Abidjan, $0.9998$ in Kigali, $\texttt{MSE} < 0.0014~^\circ C ^2$), while $\texttt{TKAN}$ variants minimize absolute errors for precipitation forecasting in low-rainfall regimes. The customized $\texttt{TKAN}$ models demonstrate improvements over the standard $\texttt{TKAN}$ across both datasets. Classical \texttt{RNNs} remain highly competitive for atmospheric pressure ($R^2 \approx 0.83{-}0.86$), outperforming $\texttt{KAN}$-based models in this task. These results highlight the potential of spline-based neural architectures for efficient and data-efficient forecasting.

cs.LG↗

arXiv2024

AI and the Future of Work in Africa White Paper

This white paper is the output of a multidisciplinary workshop in Nairobi (Nov 2023). Led by a cross-organisational team including Microsoft Research, NEPAD, Lelapa AI, and University of Oxford. The workshop brought together diverse thought-leaders from various sectors and backgrounds to discuss the implications of Generative AI for the future of work in Africa. Discussions centred around four key themes: Macroeconomic Impacts; Jobs, Skills and Labour Markets; Workers' Perspectives and Africa-Centris AI Platforms. The white paper provides an overview of the current state and trends of generative AI and its applications in different domains, as well as the challenges and risks associated with its adoption and regulation. It represents a diverse set of perspectives to create a set of insights and recommendations which aim to encourage debate and collaborative action towards creating a dignified future of work for everyone across Africa.

cs.HC↗

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↗

arXiv2014

Evans function and Fredholm determinants

We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first order linear differential operators on the real line. The applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations, for example reaction-diffusion or solitary wave equations. The Evans function and transmission coefficient, which are both finite determinants, are natural tools for both analytic and numerical determination of eigenvalues of such linear operators. However, inverting the eigenvalue problem by the free state operator generates a natural linear integral eigenvalue problem whose solvability is determined through the corresponding infinite Fredholm determinant. The relationship between all three determinants has received a lot of recent attention. We focus on the case when the underlying Fredholm operator is a trace class perturbation of the identity. Our new results include: (i) clarification of the sense in which the Evans function and transmission coefficient are equivalent; and (ii) proof of the equivalence of the transmission coefficient and Fredholm determinant, in particular in the case of distinct far fields.

math.AP↗

arXiv2012

Numerical evaluation of operator determinants

For any integral operator $K$ in the Schatten--von Neumann classes of compact operators and its approximated operator $K_N\sim(N\ge1)$ obtained by using for example a quadrature or projection method, we show that the convergence of the approximate $p$-modified Fredholm determinants $\sideset{}{_{Np}}\det(I_N+zK_N)$ to the $p$-modified Fredholm determinants $\sideset{}{_p}\det(I_\mathcal{H}+zK)$ is uniform for all $p\ge1$. As a result, we give the rate of convergences when evaluating at an eigenvalue or at an element of the resolvent set of $K$.

math.NA↗