The dictionary learning problem, representing data as a combination of few atoms, has long stood as a popular method for learning representations in statistics and signal processing. The most popular dictionary learning algorithm alternates between sparse coding and dictionary update steps, and a rich literature has studied its theoretical convergence. The growing popularity of neurally plausible unfolded sparse coding networks has led to the empirical finding that backpropagation through such networks performs dictionary learning. This paper offers the first theoretical proof for these empirical results through PUDLE, a Provable Unfolded Dictionary LEarning method. We highlight the impact of loss, unfolding, and backpropagation on convergence. We discover an implicit acceleration: as a function of unfolding, the backpropagated gradient converges faster and is more accurate than the gradient from alternating minimization. We complement our findings through synthetic and image denoising experiments. The findings support the use of accelerated deep learning optimizers and unfolded networks for dictionary learning.

Sparse Bayesian learning (SBL) is a powerful framework for tackling the sparse coding problem while also providing uncertainty quantification. However, the most popular inference algorithms for SBL become too expensive for high-dimensional problems due to the need to maintain a large covariance matrix. To resolve this issue, we introduce a new SBL inference algorithm that avoids explicit computation of the covariance matrix, thereby saving significant time and space. Instead of performing costly matrix inversions, our covariance-free method solves multiple linear systems to obtain provably unbiased estimates of the posterior statistics needed by SBL. These systems can be solved in parallel, enabling further acceleration of the algorithm via graphics processing units. In practice, our method can be up to thousands of times faster than existing baselines, reducing hours of computation time to seconds. We showcase how our new algorithm enables SBL to tractably tackle high-dimensional signal recovery problems, such as deconvolution of calcium imaging data and multi-contrast reconstruction of magnetic resonance images. Finally, we open-source a toolbox containing all of our implementations to drive future research in SBL.

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard $\ell_1$ minimization problem. We propose weighted $\ell_0$ and weighted $\ell_1$ metrics that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted $\ell_1$ and weighted $\ell_0$. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.

Convolutional dictionary learning (CDL), the problem of estimating shift-invariant templates from data, is typically conducted in the absence of a prior/structure on the templates. In data-scarce or low signal-to-noise ratio (SNR) regimes, which have received little attention from the community, learned templates overfit the data and lack smoothness, which can affect the predictive performance of downstream tasks. To address this limitation, we propose GPCDL, a convolutional dictionary learning framework that enforces priors on templates using Gaussian Processes (GPs). With the focus on smoothness, we show theoretically that imposing a GP prior is equivalent to Wiener filtering the learned templates, thereby suppressing high-frequency components and promoting smoothness. We show that the algorithm is a simple extension of the classical iteratively reweighted least squares, which allows the flexibility to experiment with different smoothness assumptions. Through simulation, we show that GPCDL learns smooth dictionaries with better accuracy than the unregularized alternative across a range of SNRs. Through an application to neural spiking data from rats, we show that learning templates by GPCDL results in a more accurate and visually-interpretable smooth dictionary, leading to superior predictive performance compared to non-regularized CDL, as well as parametric alternatives.

Principal component analysis, dictionary learning, and auto-encoders are all unsupervised methods for learning representations from a large amount of training data. In all these methods, the higher the dimensions of the input data, the longer it takes to learn. We introduce a class of neural networks, termed RandNet, for learning representations using compressed random measurements of data of interest, such as images. RandNet extends the convolutional recurrent sparse auto-encoder architecture to dense networks and, more importantly, to the case when the input data are compressed random measurements of the original data. Compressing the input data makes it possible to fit a larger number of batches in memory during training. Moreover, in the case of sparse measurements,training is more efficient computationally. We demonstrate that, in unsupervised settings, RandNet performs dictionary learning using compressed data. In supervised settings, we show that RandNet can classify MNIST images with minimal loss in accuracy, despite being trained with random projections of the images that result in a 50% reduction in size. Overall, our results provide a general principled framework for training neural networks using compressed data.

Filter banks are a popular tool for the analysis of piecewise smooth signals such as natural images. Motivated by the empirically observed properties of scale and detail coefficients of images in the wavelet domain, we propose a hierarchical deep generative model of piecewise smooth signals that is a recursion across scales: the low pass scale coefficients at one layer are obtained by filtering the scale coefficients at the next layer, and adding a high pass detail innovation obtained by filtering a sparse vector. This recursion describes a linear dynamic system that is a non-Gaussian Markov process across scales and is closely related to multilayer-convolutional sparse coding (ML-CSC) generative model for deep networks, except that our model allows for deeper architectures, and combines sparse and non-sparse signal representations. We propose an alternating minimization algorithm for learning the filters in this hierarchical model given observations at layer zero, e.g., natural images. The algorithm alternates between a coefficient-estimation step and a filter update step. The coefficient update step performs sparse (detail) and smooth (scale) coding and, when unfolded, leads to a deep neural network. We use MNIST to demonstrate the representation capabilities of the model, and its derived features (coefficients) for classification.

We consider the problem of learning recurring convolutional patterns from data that are not necessarily real valued, such as binary or count-valued data. We cast the problem as one of learning a convolutional dictionary, subject to sparsity constraints, given observations drawn from a distribution that belongs to the canonical exponential family. We propose two general approaches towards its solution. The first approach uses the $\ell_0$ pseudo-norm to enforce sparsity and is reminiscent of the alternating-minimization algorithm for classical convolutional dictionary learning (CDL). The second approach, which uses the $\ell_1$ norm to enforce sparsity, generalizes to the exponential family the recently-shown connection between CDL and a class of ReLU auto-encoders for Gaussian observations. The two approaches can each be interpreted as an auto-encoder, the weights of which are in one-to-one correspondence with the parameters of the convolutional dictionary. Our key insight is that, unless the observations are Gaussian valued, the input fed into the encoder ought to be modified iteratively, and in a specific manner, using the parameters of the dictionary. Compared to the $\ell_0$ approach, once trained, the forward pass through the $\ell_1$ encoder computes sparse codes orders of magnitude more efficiently. We apply the two approaches to the unsupervised learning of the stimulus effect from neural spiking data acquired in the barrel cortex of mice in response to periodic whisker deflections. We demonstrate that they are both superior to generalized linear models, which rely on hand-crafted features.

Convolutional dictionary learning (CDL) has become a popular method for learning sparse representations from data. State-of-the-art algorithms perform dictionary learning (DL) through an optimization-based alternating-minimization procedure that comprises a sparse coding and a dictionary update step respectively. Here, we draw connections between CDL and neural networks by proposing an architecture for CDL termed the constrained recurrent sparse auto-encoder (CRsAE). We leverage the interpretation of the alternating-minimization algorithm for DL as an Expectation-Maximization algorithm to develop auto-encoders (AEs) that, for the first time, enable the simultaneous training of the dictionary and regularization parameter. The forward pass of the encoder, which performs sparse coding, solves the E-step using an encoding matrix and a soft-thresholding non-linearity imposed by the FISTA algorithm. The encoder in this regard is a variant of residual and recurrent neural networks. The M-step is implemented via a two-stage back-propagation. In the first stage, we perform back-propagation through the AE formed by the encoder and a linear decoder whose parameters are tied to the encoder. This stage parallels the dictionary update step in DL. In the second stage, we update the regularization parameter by performing back-propagation through the encoder using a loss function that includes a prior on the parameter motivated by Bayesian statistics. We leverage GPUs to achieve significant computational gains relative to state-of-the-art optimization-based approaches to CDL. We apply CRsAE to spike sorting, the problem of identifying the time of occurrence of neural action potentials in recordings of electrical activity from the brain. We demonstrate on recordings lasting hours that CRsAE speeds up spike sorting by 900x compared to notoriously slow classical algorithms based on convex optimization.

In a data set comprising hundreds to thousands of neuronal time series (Brown et al., 2004), the ability to automatically identify subgroups of time series that respond similarly to an exogenous stimulus or contingency can provide insights into how neural computation is implemented at the level of groups of neurons. We consider the problem of clustering multiple time series that exhibit nonlinear dynamics into an a-priori-unknown number of subgroups. Existing model-based approaches for clustering multiple time series rely on a generative model of the time series that is a mixture of linear-Gaussian state-space models, and can be further classified according to whether the number of mixture components is assumed to be known a priori, and according to the choice of inference procedure (MCMC or variational Bayes) (Inoue et al., 2006; Chiappa and Barber, 2007; Nieto-Barajas et al., 2014; Middleton, 2014; Saad and Mansinghka, 2018). In all cases, the linear-Gaussian assumption is crucial: it enables exact evaluation of the likelihood using a Kalman filter and the ability to sample exactly from the state sequences underlying each of the time series. For nonlinear and/or non-Gaussian state-space models, this likelihood cannot be evaluated in closed form and exact sampling is not possible.

Given a convolutional dictionary underlying a set of observed signals, can a carefully designed auto-encoder recover the dictionary in the presence of noise? We introduce an auto-encoder architecture, termed constrained recurrent sparse auto-encoder (CRsAE), that answers this question in the affirmative. Given an input signal and an approximate dictionary, the encoder finds a sparse approximation using FISTA. The decoder reconstructs the signal by applying the dictionary to the output of the encoder. The encoder and decoder in CRsAE parallel the sparse-coding and dictionary update steps in optimization-based alternating-minimization schemes for dictionary learning. As such, the parameters of the encoder and decoder are not independent, a constraint which we enforce for the first time. We derive the back-propagation algorithm for CRsAE. CRsAE is a framework for blind source separation that, only knowing the number of sources (dictionary elements), and assuming sparsely-many can overlap, is able to separate them. We demonstrate its utility in the context of spike sorting, a source separation problem in computational neuroscience. We demonstrate the ability of CRsAE to recover the underlying dictionary and characterize its sensitivity as a function of SNR.