arXiv subjects

Radu Balan

Publications and source records attributed to Radu Balan.

At least 19 recordsLinked to original sources

arXiv2026

Robust Uniform Recovery of Structured Signals from Nonlinear Observations

While it is well known that the restricted isometry property (RIP) guarantees uniform sparse recovery from noisy linear measurements, uniform recovery of structured signals from nonlinear observations remains much less understood. This paper shows that the restricted approximate invertibility condition (RAIC) provides a unified approach to this end. Particularly, uniform recovery is achieved by projected gradient descent (PGD) with gradients obeying RAIC for all signals. As an application, under a large class of piecewise Lipschitz link functions (possibly discontinuous), we develop a uniform recovery theory for Gaussian single-index model by establishing the uniform RAIC for the gradient of the (scaled) $\ell_2$ loss via a covering argument. The theory generalizes the nonuniform recovery guarantees due to Plan and Vershynin (2016); Oymak and Soltanolkotabi (2017) and exhibits additional error terms that can be interpreted as the cost of uniform recovery. Intriguingly, in the three canonical settings of (a) sparse recovery via PGD with $\ell_0$ projection (i.e., iterative hard thresholding (IHT)), (b) sparse recovery via PGD with $\ell_1$ projection, and (c) recovering approximately sparse signals via PGD with $\ell_1$ projection, the additional error terms are negligible and in turn our uniform recovery error rates are at the same order of existing nonuniform ones, up to log factors. Our results hence improve on Genzel and Stollenwerk (2023). Under the specific nonlinearity of 1-bit quantization, we use a VC dimension argument to show that the uniform recovery error of IHT is at the same order of the nonuniform recovery error, with no loss of log factor. In addition, we show that the robustness of PGD to noise and corruption can be incorporated elegantly by bounding a single additional random process that captures the gradient mismatch.

cs.IT↗

arXiv2025

ScoresActivation: A New Activation Function for Model Agnostic Global Explainability by Design

Understanding the decision of large deep learning models is a critical challenge for building transparent and trustworthy systems. Although the current post hoc explanation methods offer valuable insights into feature importance, they are inherently disconnected from the model training process, limiting their faithfulness and utility. In this work, we introduce a novel differentiable approach to global explainability by design, integrating feature importance estimation directly into model training. Central to our method is the ScoresActivation function, a feature-ranking mechanism embedded within the learning pipeline. This integration enables models to prioritize features according to their contribution to predictive performance in a differentiable and end-to-end trainable manner. Evaluations across benchmark datasets show that our approach yields globally faithful, stable feature rankings aligned with SHAP values and ground-truth feature importance, while maintaining high predictive performance. Moreover, feature scoring is 150 times faster than the classical SHAP method, requiring only 2 seconds during training compared to SHAP's 300 seconds for feature ranking in the same configuration. Our method also improves classification accuracy by 11.24% with 10 features (5 relevant) and 29.33% with 16 features (5 relevant, 11 irrelevant), demonstrating robustness to irrelevant inputs. This work bridges the gap between model accuracy and interpretability, offering a scalable framework for inherently explainable machine learning.

cs.LG↗

arXiv2025

Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings

We study permutation-invariant embeddings of $d$-dimensional point sets, which are defined by sorting $D$ independent one-dimensional projections of the input. Such embeddings arise in graph deep learning where outputs should be invariant to permutations of graph nodes. Previous work showed that for large enough $D$ and projections in general position, this mapping is injective, and moreover satisfies a bi-Lipschitz condition. However, two gaps remain: firstly, the optimal size $D$ required for injectivity is not yet known, and secondly, no estimates of the bi-Lipschitz constants of the mapping are known. In this paper, we make substantial progress in addressing both of these gaps. Regarding the first gap, we improve upon the best known upper bounds for the embedding dimension $D$ necessary for injectivity, and also provide a lower bound on the minimal injectivity dimension. Regarding the second gap, we construct matrices of projection vectors, so that the bi-Lipschitz distortion of the mapping depends quadratically on the number of points $n$, and is completely independent of the dimension $d$. We also show that for any choice of projection vectors, the distortion of the mapping will never be better than a bound proportional to the square root of $n$. Finally, we show that similar guarantees can be provided even when linear projections are applied to the mapping to reduce its dimension.

cs.LG↗

arXiv2025

PICore: Physics-Informed Unsupervised Coreset Selection for Data Efficient Neural Operator Training

Neural operators offer a powerful paradigm for solving partial differential equations (PDEs) that cannot be solved analytically by learning mappings between function spaces. However, there are two main bottlenecks in training neural operators: they require a significant amount of training data to learn these mappings, and this data needs to be labeled, which can only be accessed via expensive simulations with numerical solvers. To alleviate both of these issues simultaneously, we propose PICore, an unsupervised coreset selection framework that identifies the most informative training samples without requiring access to ground-truth PDE solutions. PICore leverages a physics-informed loss to select unlabeled inputs by their potential contribution to operator learning. After selecting a compact subset of inputs, only those samples are simulated using numerical solvers to generate labels, reducing annotation costs. We then train the neural operator on the reduced labeled dataset, significantly decreasing training time as well. Across four diverse PDE benchmarks and multiple coreset selection strategies, PICore achieves up to 78% average increase in training efficiency relative to supervised coreset selection methods with minimal changes in accuracy. We provide code at https://github.com/Asatheesh6561/PICore.

cs.LG↗

arXiv2025

CUR Matrix Approximation through Convex Optimization for Feature Selection

The singular value decomposition (SVD) is commonly used in applications requiring a low rank matrix approximation. However, the singular vectors cannot be interpreted in terms of the original data. For applications requiring this type of interpretation, e.g., selection of important data matrix columns or rows, the approximate CUR matrix factorization can be used. Work on the CUR matrix approximation has generally focused on algorithm development, theoretical guarantees, and applications. In this work, we present a novel deterministic CUR formulation and algorithm with theoretical convergence guarantees. The algorithm utilizes convex optimization, finds important columns and rows separately, and allows the user to control the number of important columns and rows selected from the original data matrix. We present numerical results and demonstrate the effectiveness of our CUR algorithm as a feature selection method on gene expression data. These results are compared to those using the SVD and other CUR algorithms as the feature selection method. Lastly, we present a novel application of CUR as a feature selection method to determine discriminant proteins when clustering protein expression data in a self-organizing map (SOM), and compare the performance of multiple CUR algorithms in this application.

math.NA↗

arXiv2024

Stability of sorting based embeddings

Consider a group $G$ of order $M$ acting unitarily on a real inner product space $V$. We show that the sorting based embedding obtained by applying a general linear map $\alpha : \mathbb{R}^{M \times N} \to \mathbb{R}^D$ to the invariant map $\beta_\Phi : V \to \mathbb{R}^{M \times N}$ given by sorting the coorbits $(\langle v, g \phi_i \rangle_V)_{g \in G}$, where $(\phi_i)_{i=1}^N \in V$, satisfies a bi-Lipschitz condition if and only if it separates orbits. Additionally, we note that any invariant Lipschitz continuous map (into a Hilbert space) factors through the sorting based embedding, and that any invariant continuous map (into a locally convex space) factors through the sorting based embedding as well.

math.FA↗

arXiv2024

Factorization of positive-semidefinite operators with absolutely summable entries

A problem by Feichtinger, Heil, and Larson asks whether every infinite matrix $A$ with $\sum_{k,l}|A_{kl}| < \infty$ (an equivalent substitute for the Feichtinger algebra) that is positive-semidefinite admits a symmetric rank-one decomposition $A = \sum_k f_k^*\otimes f_k$ with $\sum_k \|f_k\|_{1}^2 < \infty$. In the finite-dimensional setting, we analyze the corresponding quantitative $\ell_1^n$ optimization problem by an exact reformulation as a linear program over measures, derive its dual, and prove strong duality. We then obtain an equivalent adjoint formulation regarding the quality of a convex relaxation. In the infinite-dimensional setting, we first provide a negative answer to this question using a concurrent finite-dimensional result by Bandeira-Mixon-Steinerberger. We further study the collection of operators for which such decomposition exists, showing that they are dense in a suitable topology and invariant under the action of the positive-coefficient analytic Wiener subalgebra. In addition, we give a sufficient condition for successful rank-one decomposition in terms of $2$-summing factorization, and we characterize exactly when $A^{1/2}$ is $2$-summing.

math.FA↗

arXiv2024

Approximation of the Proximal Operator of the $\ell_\infty$ Norm Using a Neural Network

Computing the proximal operator of the $\ell_\infty$ norm, $\textbf{prox}_{\alpha ||\cdot||_\infty}(\mathbf{x})$, generally requires a sort of the input data, or at least a partial sort similar to quicksort. In order to avoid using a sort, we present an $O(m)$ approximation of $\textbf{prox}_{\alpha ||\cdot||_\infty}(\mathbf{x})$ using a neural network. A novel aspect of the network is that it is able to accept vectors of varying lengths due to a feature selection process that uses moments of the input data. We present results on the accuracy of the approximation, feature importance, and computational efficiency of the approach. We show that the network outperforms a "vanilla neural network" that does not use feature selection. We also present an algorithm with corresponding theory to calculate $\textbf{prox}_{\alpha ||\cdot||_\infty}(\mathbf{x})$ exactly, relate it to the Moreau decomposition, and compare its computational efficiency to that of the approximation.

math.NA↗

arXiv2023

G-Invariant Representations using Coorbits: Injectivity Properties

Consider a finite-dimensional real vector space equipped with a finite group acting unitarily on it. We address the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our approach is based on subsets of sorted coorbits with respect to selected window vectors. We derive conditions under which such embeddings are injective, using techniques from semi-algebraic analysis. These results are then applied to the specific case of planar rotation invariance.

math.RT↗

arXiv2023

G-Invariant Representations using Coorbits: Bi-Lipschitz Properties

Consider a finite dimensional real vector space and a finite group acting unitarily on it. We study the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding is based on subsets of sorted coorbits. Our main result shows that, whenever such embeddings are injective, they are automatically bi-Lipschitz. Additionally, we demonstrate that stable embeddings can be achieved with reduced dimensionality, and that any continuous or Lipschitz $G$-invariant map can be factorized through these embeddings.

math.RT↗

arXiv2023

Relationships between the Phase Retrieval Problem and Permutation Invariant Embeddings

This paper discusses the connection between the phase retrieval problem and permutation invariant embeddings. We show that the real phase retrieval problem for $\mathbb{R}^d/O(1)$ is equivalent to Euclidean embeddings of the quotient space $\mathbb{R}^{2\times d}/S_2$ performed by the sorting encoder introduced in an earlier work. In addition, this relationship provides us with inversion algorithms of the orbits induced by the group of permutation matrices.

math.FA↗

arXiv2023

Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations

Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a \textit{coupled multiwavelets neural operator} (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a $2\times \sim 4\times$ improvement relative $L$2 error compared to the best results from the state-of-the-art models.

cs.LG↗

arXiv2022

Motion correction in MRI using deep learning and a novel hybrid loss function

Purpose To develop and evaluate a deep learning-based method (MC-Net) to suppress motion artifacts in brain magnetic resonance imaging (MRI). Methods MC-Net was derived from a UNet combined with a two-stage multi-loss function. T1-weighted axial brain images contaminated with synthetic motions were used to train the network. Evaluation used simulated T1 and T2-weighted axial, coronal, and sagittal images unseen during training, as well as T1-weighted images with motion artifacts from real scans. Performance indices included the peak signal to noise ratio (PSNR), structural similarity index measure (SSIM), and visual reading scores. Two clinical readers scored the images. Results The MC-Net outperformed other methods implemented in terms of PSNR and SSIM on the T1 axial test set. The MC-Net significantly improved the quality of all T1-weighted images (for all directions and for simulated as well as real motion artifacts), both on quantitative measures and visual scores. However, the MC-Net performed poorly on images of untrained contrast (T2-weighted). Conclusion The proposed two-stage multi-loss MC-Net can effectively suppress motion artifacts in brain MRI without compromising image context. Given the efficiency of the MC-Net (single image processing time ~40ms), it can potentially be used in real clinical settings. To facilitate further research, the code and trained model are available at https://github.com/MRIMoCo/DL_Motion_Correction.

eess.IV↗

arXiv2022

VQ-Flows: Vector Quantized Local Normalizing Flows

Normalizing flows provide an elegant approach to generative modeling that allows for efficient sampling and exact density evaluation of unknown data distributions. However, current techniques have significant limitations in their expressivity when the data distribution is supported on a low-dimensional manifold or has a non-trivial topology. We introduce a novel statistical framework for learning a mixture of local normalizing flows as "chart maps" over the data manifold. Our framework augments the expressivity of recent approaches while preserving the signature property of normalizing flows, that they admit exact density evaluation. We learn a suitable atlas of charts for the data manifold via a vector quantized auto-encoder (VQ-AE) and the distributions over them using a conditional flow. We validate experimentally that our probabilistic framework enables existing approaches to better model data distributions over complex manifolds.

cs.LG↗

arXiv2022

Permutation Invariant Representations with Applications to Graph Deep Learning

This paper presents primarily two Euclidean embeddings of the quotient space generated by matrices that are identified modulo arbitrary row permutations. The original application is in deep learning on graphs where the learning task is invariant to node relabeling. Two embedding schemes are introduced, one based on sorting and the other based on algebras of multivariate polynomials. While both embeddings exhibit a computational complexity exponential in problem size, the sorting based embedding is globally bi-Lipschitz and admits a low dimensional target space. Additionally, an almost everywhere injective scheme can be implemented with minimal redundancy and low computational cost. In turn, this proves that almost any classifier can be implemented with an arbitrary small loss of performance. Numerical experiments are carried out on two data sets, a chemical compound data set (QM9) and a proteins data set (PROTEINS).

math.FA↗

arXiv2021

Convergence Guarantees for Deep Epsilon Greedy Policy Learning

Policy learning is a quickly growing area. As robotics and computers control day-to-day life, their error rate needs to be minimized and controlled. There are many policy learning methods and bandit methods with provable error rates that accompany them. We show an error or regret bound and convergence of the Deep Epsilon Greedy method which chooses actions with a neural network's prediction. We also show that Epsilon Greedy method regret upper bound is minimized with cubic root exploration. In experiments with the real-world dataset MNIST, we construct a nonlinear reinforcement learning problem. We witness how with either high or low noise, some methods do and some do not converge which agrees with our proof of convergence.

cs.LG↗

arXiv2021

Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames

The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames is natural in the sense that it corresponds to quantum tomography of impure states. We provide computable global stability bounds for the quasi-linear analysis map $\beta$ and a path forward for understanding related problems in terms of the differential geometry of key spaces. In particular, we manifest a Whitney stratification of the positive semidefinite matrices of low rank which allows us to ``stratify'' the computation of the global stability bound. We show that for the impure state case no such global stability bounds can be obtained for the non-linear analysis map $\alpha$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $\beta$ analysis map provides novel conditions for a frame to be generalized phase retrievable.

quant-ph↗

arXiv2021

An Exact Hypergraph Matching Algorithm for Nuclear Identification in Embryonic Caenorhabditis elegans

Finding an optimal correspondence between point sets is a common task in computer vision. Existing techniques assume relatively simple relationships among points and do not guarantee an optimal match. We introduce an algorithm capable of exactly solving point set matching by modeling the task as hypergraph matching. The algorithm extends the classical branch and bound paradigm to select and aggregate vertices under a proposed decomposition of the multilinear objective function. The methodology is motivated by Caenorhabditis elegans, a model organism used frequently in developmental biology and neurobiology. The embryonic C. elegans contains seam cells that can act as fiducial markers allowing the identification of other nuclei during embryo development. The proposed algorithm identifies seam cells more accurately than established point-set matching methods, while providing a framework to approach other similarly complex point set matching tasks.

cs.CV↗