Babhru Joshi
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Babhru Joshi has a total of 11 co-authors. Here are the top co-authors by number of shared publications:
- Aaron Berk
Department of Mathematics & Statistics, McGill University, Montréal, QC, Canada
1 shared publication - Ali Ahmed
Information Technology University Lahore Pakistan
1 shared publication - Alireza Aghasi
Robinson College of Business, Georgia State University, Atlanta, GA, USA
1 shared publication - Halyun Jeong
Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095 USA, and Department of Mathematics & Statistics, The State University of New York at Albany, Albany, NY 12222 USA
1 shared publication - Matthew S. Scott
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
1 shared publication - Michael P. Friedlander
University of British Columbia
1 shared publication
Latest Publications
- 8 November 2022
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Journal on Selected Areas in Information Theory
- Vol. 3 (3) , 502-512
A Coherence Parameter Characterizing Generative Compressed Sensing With Fourier Measurements
In 1, a mathematical framework was developed for compressed sensing guarantees in the setting where the measurement matrix is Gaussian and the signal structure is the range of a generative neural network (GNN). The problem of compressed sensing with GNNs has since been extensively analyzed when the measurement matrix and/or network weights follow a subgaussian distribution. We move beyond the subgaussian assumption, to measurement matrices that are derived by sampling uniformly at random rows of a unitary matrix (including subsampled Fourier measurements as a special case). Specifically, we prove the first known restricted isometry guarantee for generative compressed sensing (GCS) with subsampled isometries, and provide recovery bounds with nearly order-optimal sample complexity, addressing an open problem of 2, p. 10. Recovery efficacy is characterized by the coherence, a new parameter, which measures the interplay between the range of the network and the measurement matrix. Our approach relies on subspace counting arguments and ideas central to high-dimensional probability. Furthermore, we propose a regularization strategy for training GNNs to have favourable coherence with the measurement operator. We provide compelling numerical simulations that support this regularized training strategy: our strategy yields low coherence networks that require fewer measurements for signal recovery. This, together with our theoretical results, supports coherence as a natural quantity for characterizing GCS with subsampled isometries.
- 8 November 2022
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Journal on Selected Areas in Information Theory
- Vol. 3 (3) , 502-512
A Coherence Parameter Characterizing Generative Compressed Sensing With Fourier Measurements
In 1, a mathematical framework was developed for compressed sensing guarantees in the setting where the measurement matrix is Gaussian and the signal structure is the range of a generative neural network (GNN). The problem of compressed sensing with GNNs has since been extensively analyzed when the measurement matrix and/or network weights follow a subgaussian distribution. We move beyond the subgaussian assumption, to measurement matrices that are derived by sampling uniformly at random rows of a unitary matrix (including subsampled Fourier measurements as a special case). Specifically, we prove the first known restricted isometry guarantee for generative compressed sensing (GCS) with subsampled isometries, and provide recovery bounds with nearly order-optimal sample complexity, addressing an open problem of 2, p. 10. Recovery efficacy is characterized by the coherence, a new parameter, which measures the interplay between the range of the network and the measurement matrix. Our approach relies on subspace counting arguments and ideas central to high-dimensional probability. Furthermore, we propose a regularization strategy for training GNNs to have favourable coherence with the measurement operator. We provide compelling numerical simulations that support this regularized training strategy: our strategy yields low coherence networks that require fewer measurements for signal recovery. This, together with our theoretical results, supports coherence as a natural quantity for characterizing GCS with subsampled isometries.
- 27 May 2022
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Signal Processing
- Vol. 70 (1053587X) , 2713-2727
Polar Deconvolution of Mixed Signals
The signal demixing problem seeks to separate a superposition of multiple signals into its constituent components. This paper studies a two-stage approach that first decompresses and subsequently deconvolves the noisy and undersampled observations of the superposition using two convex programs. Probabilistic error bounds are given on the accuracy with which this process approximates the individual signals. The theory of polar convolution of convex sets and gauge functions plays a central role in the analysis and solution process. If the measurements are random and the noise is bounded, this approach stably recovers low-complexity and mutually incoherent signals, with high probability and with near optimal sample complexity. We develop an efficient algorithm, based on level-set and conditional-gradient methods, that solves the convex optimization problems with sublinear iteration complexity and linear space requirements. Numerical experiments on both real and synthetic data confirm the theory and the efficiency of the approach.
- 27 May 2022
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Signal Processing
- Vol. 70 (1053587X) , 2713-2727
Polar Deconvolution of Mixed Signals
The signal demixing problem seeks to separate a superposition of multiple signals into its constituent components. This paper studies a two-stage approach that first decompresses and subsequently deconvolves the noisy and undersampled observations of the superposition using two convex programs. Probabilistic error bounds are given on the accuracy with which this process approximates the individual signals. The theory of polar convolution of convex sets and gauge functions plays a central role in the analysis and solution process. If the measurements are random and the noise is bounded, this approach stably recovers low-complexity and mutually incoherent signals, with high probability and with near optimal sample complexity. We develop an efficient algorithm, based on level-set and conditional-gradient methods, that solves the convex optimization problems with sublinear iteration complexity and linear space requirements. Numerical experiments on both real and synthetic data confirm the theory and the efficiency of the approach.
- 14 September 2020
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Signal Processing
- Vol. 68 (1053587X) , 6366-6379
Bilinear Compressed Sensing Under Known Signs via Convex Programming
We consider the bilinear inverse problem of recovering two vectors, $\vx\in\R^L$ and $\vw\in\R^L$ , from their entrywise product. We consider the case where $\vx$ and $\vw$ have known signs and are sparse with respect to known dictionaries of size $K$ and $N$ , respectively. Here, $K$ and $N$ may be larger than, smaller than, or equal to $L$ . We introduce $\ell_1$ -BranchHull, which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved. Under the assumptions that $\vx$ and $\vw$ satisfy a comparable-effective-sparsity condition and are $S_1$ - and $S_2$ -sparse with respect to a random dictionary, we present a recovery guarantee in a noisy case. We show that $\ell_1$ -BranchHull is robust to small dense noise with high probability if the number of measurements satisfy $L\geq\Omega\left((S_1+S_2)\log^{2}(K+N)\right)$ . Numerical experiments show that the scaling constant in the theorem is not too large. We also introduce variants of $\ell_1$ -BranchHull for the purposes of tolerating noise and outliers, and for the purpose of recovering piecewise constant signals. We provide an ADMM implementation of these variants and show they can extract piecewise constant behavior from real images.- 14 September 2020
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Signal Processing
- Vol. 68 (1053587X) , 6366-6379
Bilinear Compressed Sensing Under Known Signs via Convex Programming
We consider the bilinear inverse problem of recovering two vectors, $\vx\in\R^L$ and $\vw\in\R^L$ , from their entrywise product. We consider the case where $\vx$ and $\vw$ have known signs and are sparse with respect to known dictionaries of size $K$ and $N$ , respectively. Here, $K$ and $N$ may be larger than, smaller than, or equal to $L$ . We introduce $\ell_1$ -BranchHull, which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved. Under the assumptions that $\vx$ and $\vw$ satisfy a comparable-effective-sparsity condition and are $S_1$ - and $S_2$ -sparse with respect to a random dictionary, we present a recovery guarantee in a noisy case. We show that $\ell_1$ -BranchHull is robust to small dense noise with high probability if the number of measurements satisfy $L\geq\Omega\left((S_1+S_2)\log^{2}(K+N)\right)$ . Numerical experiments show that the scaling constant in the theorem is not too large. We also introduce variants of $\ell_1$ -BranchHull for the purposes of tolerating noise and outliers, and for the purpose of recovering piecewise constant signals. We provide an ADMM implementation of these variants and show they can extract piecewise constant behavior from real images.