Deep learning improves solving medical imaging problems with sparse data.
problem Solving underdetermined inverse problems in medical imaging.
method Analyzing the structure of training data suitable for deep learning to solve highly non-linear underdetermined systems.
result Deep learning can learn reconstruction maps from training data for highly underdetermined systems.
Bayesian model tackles spatio-temporal underdetermined problems.
problem Solving underdetermined linear inverse problems with spatial and temporal sparsity constraints.
method Generalized spike-and-slab prior with transformed Gaussian process, expectation propagation algorithm, and approximations for scalability.
result Demonstrated effectiveness on synthetic and real data sets.
Improved audio source separation for underdetermined conditions.
problem Underdetermined source separation challenges for non-NMF-compliant sound sources.
method Generalized Multichannel Variational Autoencoder (GMVAE) that extends Conditional VAE for underdetermined cases.
result The GMVAE method outperformed MNMF in underdetermined source separation tasks.
Half-AVAE enhances VAE for underdetermined ICA with adversarial training.
problem Challenges in ICA under underdetermined conditions.
method Encoder-free VAE with adversarial networks and EE terms.
result Half-AVAE outperforms baseline models in underdetermined ICA.
This paper tackles traffic volume estimation challenges with a deep learning method.
problem Underdetermined and non-equilibrium traffic flows.
method Graph-based deep learning method with adaptive attention mechanisms.
result The proposed model achieves high accuracy even with low sensor coverage.
New techniques for compressive sensing without noise or signal statistics.
problem Support recovery in underdetermined linear regression models without prior noise and signal statistics.
method Proposes RRM and RRTA to operate OMP algorithm without noise variance or signal sparsity knowledge.
result Establishes high SNR consistency for OMP without prior noise and signal statistics.
We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth soluti…
Proposes a global model from local sparsity assumptions in inverse problems.
problem Inability of traditional sparse modeling to model entire global signals optimally.
method Constructs a global model from local sparsity assumptions using constrained underdetermined systems and ADMM optimization.
result Shows unique and stable recovery conditions for global signal representation.
The paper argues for prioritizing identifying structure over complex models for scientific discovery.
problem Underdetermination of mechanisms in high-dimensional data, leading to unreliable explanations.
method Proposes concrete standards for 'mechanistic ML' to avoid collapsing explanations.
result Large language models (LLMs) can collapse large equivalence classes of explanations, making it hard to distinguish between mechanisms.
A new algorithm selects features efficiently for sparse linear regression.
problem Finding sparse solutions to underdetermined linear systems.
method Generalized Orthogonal Least-Squares (OLS) algorithm.
result The generalized OLS algorithm outperforms existing methods in efficiency and performance.
The paper studies sparse solutions for underdetermined systems of linear equations.
problem Finding sparse solutions for noisy underdetermined systems of linear equations.
method Unified optimization model and various properties analysis.
result Upper bounds and boundary conditions for optimal solutions.
A new NMF variant tackles underdetermined problems with sparse and separable assumptions.
problem Underdetermined blind source separation, especially multispectral image unmixing.
method Sparse Separable Nonnegative Matrix Factorization (SSNMF) combining separability and sparsity assumptions. Algorithm based on SNPA and sparse nonnegative least squares.
result In noiseless settings, the algorithm recovers true underlying sources.
Study compares L1 and VG sparsity priors in inverse problems.
problem Sparse regularization in inverse problems with incomplete or corrupted measurements.
method Compared L1 regularization with Variational Garrote (VG), a probabilistic method approximating L0 sparsity.
result VG often achieves lower minimum generalization error and improved stability in strongly underdetermined regimes.
Finding sparse solutions of underdetermined systems of linear equations is a fundamental problem in signal processing and statistics which has become a subject of interest in recent years. In general, these systems have infinitely many solutions. However, it may be shown that sufficiently sparse solutions may be identi…
Double descent in condition number shows peak at equal dimensions.
problem Estimating function from random measurements.
method Analyzing the condition number of random matrices.
result Condition number is highest when d=n. A-DLISTA and VLISTA learn dictionaries and sparse representations under varying sensing matrices.
problem Learning dictionaries and sparse representations under varying sensing matrices.
method Augmented Dictionary Learning ISTA (A-DLISTA) and Variational Learning ISTA (VLISTA).
result VLISTA provides a probabilistic way to jointly learn the dictionary distribution and the reconstruction algorithm.
We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.
When solving data analysis problems it is important to integrate prior knowledge and/or structural invariances. This paper contributes by a novel framework for incorporating algebraic invariance structure into kernels. In particular, we show that algebraic properties such as sign symmetries in data, phase independence,…
Study how optimization methods' choices affect the solutions they find.
problem Characterize the solutions found by optimization methods under different potentials and norms.
method Examined mirror descent, natural gradient descent, and steepest descent for underdetermined linear regression and separable linear classification.
result The specific global minimum reached by an algorithm can be characterized by the potential or norm of the optimization geometry, independent of hyperparameters.
Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear sub-manifold of a high dimensional ambient space. We introduce SPIN, a first order projected gradient method to recover the signal components. Despite the…
Sparse elasticity reconstruction from local displacements reduces error.
problem Reconstructing elasticity from limited data.
method Sparse elasticity reconstruction theory, local clustering, alternating optimization.
result Higher spatial resolution elasticity distribution estimation.
New method constructs solution operators for PDEs with prescribed support properties.
problem Constructing solution operators for under/overdetermined PDEs with specific support properties.
method Using a recovery on curves condition and taking smooth averages over curves, we obtain integral solution operators and representation formulas.
result Our method leads to integral representation formulas for overdetermined PDEs and solution operators for underdetermined PDEs.
Unified geometric approach to image reconstruction from incomplete data.
problem Reconstruction of hidden structures from incomplete data.
method Geometric decomposition of configuration spaces into invariant foliations and moment maps, combining Vaisman and Neifeld's insights.
result Noise-resistant framework for robust computational reconstruction in imaging and structural analysis.
GEMSS discovers multiple sparse solutions in high-dimensional data.
problem Identifying multiple sparse feature combinations in high-dimensional, underdetermined systems.
method GEMSS (Gaussian Ensemble for Multiple Sparse Solutions) uses a structured spike-and-slab prior, mixture of Gaussians, and Jaccard-based penalty to optimize a single objective function via stochastic gradient descent.
result GEMSS consistently outperforms five feature selection methods on 128 experiments and real-world datasets.
This work proposes optimal decision rules for hierarchical classifiers to better align with evaluation metrics.
problem Heuristic decision rules in hierarchical classification do not align with evaluation metrics.
method Derives optimal decision rules for various prediction settings, focusing on hierarchical hFβ scores. result Optimal decision rules enhance the performance and reliability of hierarchical classifiers.
Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure for finding sparse solutions of underdetermined linear systems. This method has been shown to have strong theoretical guarantee and impressive numerical performance. In this paper, we generalize HTP from compressive sensing to a generic problem …
Recent results in Compressive Sensing have shown that, under certain conditions, the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a …
Gradient descent on matrix factorization leads to minimum nuclear norm solution.
problem Optimizing underdetermined quadratic objectives over matrices.
method Gradient descent on a full dimensional factorization of the matrix.
result Gradient descent converges to the minimum nuclear norm solution.
We give a geometric interpretation of all the m-th elliptic integrable systems associated to a k′-symmetric space N=G/G0 (in the sense of C.L. Terng). It turns out that we have to introduce the integer mk′ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the prim…
Optimal ridge penalty can be negative or zero in high-dimensional data.
problem Overfitting in high-dimensional underdetermined linear regression.
method Simulations and real-life data analysis with minimum-norm estimator.
result Optimal ridge penalty can be negative, contradicting conventional wisdom.
New method regularizes MEG inverse problem for more accurate brain activity reconstruction.
problem Underdetermined inverse problem in MEG for precise brain activity reconstruction.
method Regularization using space-time separable Gaussian process model.
result Efficient and general Bayesian source reconstruction approach demonstrated.
We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the s…
A new method for reconstructing flows from perturbed distributions.
problem Reconstructing flows from perturbed probability distributions.
method Integrable vector fields and Green's functions.
result A nonparametric flow can be computed to generate samples from a perturbed distribution.
We address the sparse signal recovery problem in the context of multiple measurement vectors (MMV) when elements in each nonzero row of the solution matrix are temporally correlated. Existing algorithms do not consider such temporal correlations and thus their performance degrades significantly with the correlations. I…
We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing spars…
BP fails to find sparsest solution for structured matrices.
problem Finding sparsest solution to linear equations with structured matrices.
method Introduced class of structured matrices for BP failure.
result Determines columns corresponding to unrecoverable non-zero entries.
Proposes using equivariant generative models for compressed sensing with unknown orientations.
problem Recovering signals with unknown orientations from underdetermined systems of linear measurements.
method Equivariant variational autoencoder as a generative prior for compressed sensing.
result Signals with unknown orientations can be recovered using iterative gradient descent on the latent space of equivariant models.
We use matricial free energy to regularize autoencoders, producing Gaussian-like codes.
problem Generating Gaussian-like codes for autoencoders.
method Define a differentiable loss function based on singular values of the code matrix, minimizing matricial free energy.
result Minimizing matricial free energy results in Gaussian-like codes that generalize.
MimicGAN learns to recover images from corrupted data without supervision.
problem Recovering images from corrupted data without known prior information.
method Generative adversarial networks (GANs) to learn corruption mimicking.
result MimicGAN outperforms existing techniques in blind image recovery and adversarial defense.
Study optimal investment under imitation of decision-changing rates.
problem Optimal investment under imitation of decision-changing rates.
method Proposed integral disparity to quantify imitation, derived general solution using variational method, analyzed asymptotic properties, validated with real data.
result Investor's optimal decisions under imitation of decision-changing rates.
The performance of sparse signal recovery from noise corrupted, underdetermined measurements can be improved if both sparsity and correlation structure of signals are exploited. One typical correlation structure is the intra-block correlation in block sparse signals. To exploit this structure, a framework, called block…
New tensor formulation reveals gradient flow's bias in linear neural networks.
problem Understanding implicit bias in linear neural network training.
method Tensor formulation of neural networks, including fully-connected, diagonal, and convolutional networks.
result Gradient flow on linear tensor networks converges to solutions of specific optimization problems.
Bayesian EP solves CS problems more accurately than other methods.
problem Finding sparse solutions to underdetermined linear systems with constraints.
method Bayesian inference with Expectation Propagation (EP) for marginal distribution computation.
result EP outperforms other methods in solving CS problems with correlated sensing matrices.
Gradient descent with specific initialization and step size achieves optimal sparse signal recovery.
problem Reconstructing a sparse signal from underdetermined linear measurements.
method Gradient descent with specific initialization, step size, and stopping time.
result Achieves the minimax rate with poly-logarithmic factors and adapts to instance difficulty.
New method separates audio sources without needing known decompositions.
problem Difficulty in training source separation models on real-world mixtures.
method Generates estimated decompositions from stereo mixtures and trains a deep learning model.
result Trained model can separate single-channel audio sources effectively.
New model improves matrix completion under noisy conditions.
problem Noisy matrix completion with structured matrices.
method Introduces permutation-rank model, analyzes SVD algorithm.
result Minimax rates equivalent to low-rank model, consistent estimator.
To recover a sparse signal from an underdetermined system, we often solve a constrained L1-norm minimization problem. In many cases, the signal sparsity and the recovery performance can be further improved by replacing the L1 norm with a "weighted" L1 norm. Without any prior information about nonzero elements of the si…
Causal deep learning tackles causal inference using tensor factor analysis.
problem Addressing causal questions in data using neural networks.
method Tensor factor analysis and neural network architectures (causal capsules, tensor transformer, multilinear projection algorithm).
result Derives deep neural networks for causal inference with tensor factor analysis.