Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary. In a widely used algorithm, Extended Dynamic Mode Decomposition, the dictionary functions are drawn from a fixed class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators. Part 2 of this paper will extend this explanation by demonstrating the subspace invariant of heterogeneous dictionaries and presenting a head-to-head numerical comparison of deepDMD and low-parameter heterogeneous dictionary learning.
Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of associative memory and provide a detailed theoretical analysis of its performance, showing that decimation is able to factorize extensive-rank matrices and to denoise them efficiently. We introduce a decimation algorithm based on ground-state search of the neural network, which shows performances that match the theoretical prediction.
Understanding pedestrian behavior patterns is a key component to building autonomous agents that can navigate among humans. We seek a learned dictionary of pedestrian behavior to obtain a semantic description of pedestrian trajectories. Supervised methods for dictionary learning are impractical since pedestrian behaviors may be unknown a priori and the process of manually generating behavior labels is prohibitively time consuming. We instead utilize a novel, unsupervised framework to create a taxonomy of pedestrian behavior observed in a specific space. First, we learn a trajectory latent space that enables unsupervised clustering to create an interpretable pedestrian behavior dictionary. We show the utility of this dictionary for building pedestrian behavior maps to visualize space usage patterns and for computing the distributions of behaviors. We demonstrate a simple but effective trajectory prediction by conditioning on these behavior labels. While many trajectory analysis methods rely on RNNs or transformers, we develop a lightweight, low-parameter approach and show results comparable to SOTA on the ETH and UCY datasets.
Extremely large-scale massive MIMO (XL-MIMO) has been reviewed as a promising technology for future wireless communications. The deployment of XL-MIMO, especially at high-frequency bands, leads to users being located in the near-field region instead of the conventional far-field. This letter proposes efficient model-based deep learning algorithms for estimating the near-field wireless channel of XL-MIMO communications. In particular, we first formulate the XL-MIMO near-field channel estimation task as a compressed sensing problem using the spatial gridding-based sparsifying dictionary, and then solve the resulting problem by applying the Learning Iterative Shrinkage and Thresholding Algorithm (LISTA). Due to the near-field characteristic, the spatial gridding-based sparsifying dictionary may result in low channel estimation accuracy and a heavy computational burden. To address this issue, we further propose a new sparsifying dictionary learning-LISTA (SDL-LISTA) algorithm that formulates the sparsifying dictionary as a neural network layer and embeds it into LISTA neural network. The numerical results show that our proposed algorithms outperform non-learning benchmark schemes, and SDL-LISTA achieves better performance than LISTA with ten times atoms reduction.
Independent component analysis (ICA) of multi-subject functional magnetic resonance imaging (fMRI) data has proven useful in providing a fully multivariate summary that can be used for multiple purposes. ICA can identify patterns that can discriminate between healthy controls (HC) and patients with various mental disorders such as schizophrenia (Sz). Temporal functional network connectivity (tFNC) obtained from ICA can effectively explain the interactions between brain networks. On the other hand, dictionary learning (DL) enables the discovery of hidden information in data using learnable basis signals through the use of sparsity. In this paper, we present a new method that leverages ICA and DL for the identification of directly interpretable patterns to discriminate between the HC and Sz groups. We use multi-subject resting-state fMRI data from $358$ subjects and form subject-specific tFNC feature vectors from ICA results. Then, we learn sparse representations of the tFNCs and introduce a new set of sparse features as well as new interpretable patterns from the learned atoms. Our experimental results show that the new representation not only leads to effective classification between HC and Sz groups using sparse features, but can also identify new interpretable patterns from the learned atoms that can help understand the complexities of mental diseases such as schizophrenia.
In this paper, we consider a sensor placement problem where sensors can move within a network over time. Sensor placement problem aims to select K sensor positions from N candidates where K < N. Most existing methods assume that sensor positions are static, i.e., they do not move, however, many mobile sensors like drones, robots, and vehicles can change their positions over time. Moreover, underlying measurement conditions could also be changed, which are difficult to cover with statically placed sensors. We tackle the problem by allowing the sensors to change their positions in their neighbors on the network. We dynamically determine the sensor positions based on graph signal sampling theory such that the non-observed signals on the network can be best recovered from the observations. For signal recovery, the dictionary is learned from a pool of observed signals. It is also used for the sensor position selection. In experiments, we validate the effectiveness of the proposed method via the mean squared error of the reconstructed signals. The proposed dynamic sensor placement outperforms the existing static ones for both synthetic and real data.
Sparse representation has attracted great attention because it can greatly save storage re- sources and find representative features of data in a low-dimensional space. As a result, it may be widely applied in engineering domains including feature extraction, compressed sensing, signal denoising, picture clustering, and dictionary learning, just to name a few. In this paper, we propose a spiking sampling network. This network is composed of spiking neurons, and it can dynamically decide which pixel points should be retained and which ones need to be masked according to the input. Our experiments demonstrate that this approach enables better sparse representation of the original image and facilitates image reconstruction compared to random sampling. We thus use this approach for compressing massive data from the dynamic vision sensor, which greatly reduces the storage requirements for event data.
With the rapid development of information technologies, centralized data processing is subject to many limitations, such as computational overheads, communication delays, and data privacy leakage. Decentralized data processing over networked terminal nodes becomes an important technology in the era of big data. Dictionary learning is a powerful representation learning method to exploit the low-dimensional structure from the high-dimensional data. By exploiting the low-dimensional structure, the storage and the processing overhead of data can be effectively reduced. In this paper, we propose a novel decentralized complete dictionary learning algorithm, which is based on $\ell^{4}$-norm maximization. Compared with existing decentralized dictionary learning algorithms, comprehensive numerical experiments show that the novel algorithm has significant advantages in terms of per-iteration computational complexity, communication cost, and convergence rate in many scenarios. Moreover, a rigorous theoretical analysis shows that the dictionaries learned by the proposed algorithm can converge to the one learned by a centralized dictionary learning algorithm at a linear rate with high probability under certain conditions.
Cryo Focused Ion-Beam Scanning Electron Microscopy (cryo FIB-SEM) enables three-dimensional and nanoscale imaging of biological specimens via a slice and view mechanism. The FIB-SEM experiments are, however, limited by a slow (typically, several hours) acquisition process and the high electron doses imposed on the beam sensitive specimen can cause damage. In this work, we present a compressive sensing variant of cryo FIB-SEM capable of reducing the operational electron dose and increasing speed. We propose two Targeted Sampling (TS) strategies that leverage the reconstructed image of the previous sample layer as a prior for designing the next subsampling mask. Our image recovery is based on a blind Bayesian dictionary learning approach, i.e., Beta Process Factor Analysis (BPFA). This method is experimentally viable due to our ultra-fast GPU-based implementation of BPFA. Simulations on artificial compressive FIB-SEM measurements validate the success of proposed methods: the operational electron dose can be reduced by up to 20 times. These methods have large implications for the cryo FIB-SEM community, in which the imaging of beam sensitive biological materials without beam damage is crucial.
Analog beamforming is the predominant approach for millimeter wave (mmWave) communication given its favorable characteristics for limited-resource devices. In this work, we aim at reducing the spectral efficiency gap between analog and digital beamforming methods. We propose a method for refined beam selection based on the estimated raw channel. The channel estimation, an underdetermined problem, is solved using compressed sensing (CS) methods leveraging angular domain sparsity of the channel. To reduce the complexity of CS methods, we propose dictionary learning iterative soft-thresholding algorithm, which jointly learns the sparsifying dictionary and signal reconstruction. We evaluate the proposed method on a realistic mmWave setup and show considerable performance improvement with respect to code-book based analog beamforming approaches.
We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function, considered in the ambient space. This class of problems finds important applications in machine learning and statistics such as the sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an $\epsilon$-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow nonsmooth objective function. Our algorithm is the first ADMM type algorithm that minimizes a nonsmooth objective over manifold -- a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method.
This paper focuses on the noiseless complete dictionary learning problem, where the goal is to represent a set of given signals as linear combinations of a small number of atoms from a learned dictionary. There are two main challenges faced by theoretical and practical studies of dictionary learning: the lack of theoretical guarantees for practically-used heuristic algorithms and their poor scalability when dealing with huge-scale datasets. Towards addressing these issues, we propose a simple and efficient algorithm that provably recovers the ground truth when applied to the nonconvex and discrete formulation of the problem in the noiseless setting. We also extend our proposed method to mini-batch and online settings where the data is huge-scale or arrives continuously over time. At the core of our proposed method lies an efficient preconditioning technique that transforms the unknown dictionary to a near-orthonormal one, for which we prove a simple alternating minimization technique converges linearly to the ground truth under minimal conditions. Our numerical experiments on synthetic and real datasets showcase the superiority of our method compared with the existing techniques.
Wasserstein dictionary learning is an unsupervised approach to learning a collection of probability distributions that generate observed distributions as Wasserstein barycentric combinations. Existing methods for Wasserstein dictionary learning optimize an objective that seeks a dictionary with sufficient representation capacity via barycentric interpolation to approximate the observed training data, but without imposing additional structural properties on the coefficients associated to the dictionary. This leads to dictionaries that densely represent the observed data, which makes interpretation of the coefficients challenging and may also lead to poor empirical performance when using the learned coefficients in downstream tasks. In contrast and motivated by sparse dictionary learning in Euclidean spaces, we propose a geometrically sparse regularizer for Wasserstein space that promotes representations of a data point using only nearby dictionary elements. We show this approach leads to sparse representations in Wasserstein space and addresses the problem of non-uniqueness of barycentric representation. Moreover, when data is generated as Wasserstein barycenters of fixed distributions, this regularizer facilitates the recovery of the generating distributions in cases that are ill-posed for unregularized Wasserstein dictionary learning. Through experimentation on synthetic and real data, we show that our geometrically regularized approach yields sparser and more interpretable dictionaries in Wasserstein space, which perform better in downstream applications.
Dictionary learning, the problem of recovering a sparsely used matrix $\mathbf{D} \in \mathbb{R}^{M \times K}$ and $N$ $s$-sparse vectors $\mathbf{x}_i \in \mathbb{R}^{K}$ from samples of the form $\mathbf{y}_i = \mathbf{D}\mathbf{x}_i$, is of increasing importance to applications in signal processing and data science. When the dictionary is known, recovery of $\mathbf{x}_i$ is possible even for sparsity linear in dimension $M$, yet to date, the only algorithms which provably succeed in the linear sparsity regime are Riemannian trust-region methods, which are limited to orthogonal dictionaries, and methods based on the sum-of-squares hierarchy, which requires super-polynomial time in order to obtain an error which decays in $M$. In this work, we introduce SPORADIC (SPectral ORAcle DICtionary Learning), an efficient spectral method on family of reweighted covariance matrices. We prove that in high enough dimensions, SPORADIC can recover overcomplete ($K > M$) dictionaries satisfying the well-known restricted isometry property (RIP) even when sparsity is linear in dimension up to logarithmic factors. Moreover, these accuracy guarantees have an ``oracle property" that the support and signs of the unknown sparse vectors $\mathbf{x}_i$ can be recovered exactly with high probability, allowing for arbitrarily close estimation of $\mathbf{D}$ with enough samples in polynomial time. To the author's knowledge, SPORADIC is the first polynomial-time algorithm which provably enjoys such convergence guarantees for overcomplete RIP matrices in the near-linear sparsity regime.
Low overhead channel estimation based on compressive sensing (CS) has been widely investigated for hybrid wideband millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems. The channel sparsifying dictionaries used in prior work are built from ideal array response vectors evaluated on discrete angles of arrival/departure. In addition, these dictionaries are assumed to be the same for all subcarriers, without considering the impacts of hardware impairments and beam squint. In this manuscript, we derive a general channel and signal model that explicitly incorporates the impacts of hardware impairments, practical pulse shaping functions, and beam squint, overcoming the limitations of mmWave MIMO channel and signal models commonly used in previous work. Then, we propose a dictionary learning (DL) algorithm to obtain the sparsifying dictionaries embedding hardware impairments, by considering the effect of beam squint without introducing it into the learning process. We also design a novel CS channel estimation algorithm under beam squint and hardware impairments, where the channel structures at different subcarriers are exploited to enable channel parameter estimation with low complexity and high accuracy. Numerical results demonstrate the effectiveness of the proposed DL and channel estimation strategy when applied to realistic mmWave channels.
Opinion summarization is the task of creating summaries capturing popular opinions from user reviews. In this paper, we introduce Geodesic Summarizer (GeoSumm), a novel system to perform unsupervised extractive opinion summarization. GeoSumm involves an encoder-decoder based representation learning model, that generates representations of text as a distribution over latent semantic units. GeoSumm generates these representations by performing dictionary learning over pre-trained text representations at multiple decoder layers. We then use these representations to quantify the relevance of review sentences using a novel approximate geodesic distance based scoring mechanism. We use the relevance scores to identify popular opinions in order to compose general and aspect-specific summaries. Our proposed model, GeoSumm, achieves state-of-the-art performance on three opinion summarization datasets. We perform additional experiments to analyze the functioning of our model and showcase the generalization ability of {\X} across different domains.
A common study area in anomaly identification is industrial images anomaly detection based on texture background. The interference of texture images and the minuteness of texture anomalies are the main reasons why many existing models fail to detect anomalies. We propose a strategy for anomaly detection that combines dictionary learning and normalizing flow based on the aforementioned questions. The two-stage anomaly detection approach already in use is enhanced by our method. In order to improve baseline method, this research add normalizing flow in representation learning and combines deep learning and dictionary learning. Improved algorithms have exceeded 95$\%$ detection accuracy on all MVTec AD texture type data after experimental validation. It shows strong robustness. The baseline method's detection accuracy for the Carpet data was 67.9%. The article was upgraded, raising the detection accuracy to 99.7%.
We consider matrix factorization (MF) with certain constraints, which finds wide applications in various areas. Leveraging variational inference (VI) and unitary approximate message passing (UAMP), we develop a Bayesian approach to MF with an efficient message passing implementation, called UAMPMF. With proper priors imposed on the factor matrices, UAMPMF can be used to solve many problems that can be formulated as MF, such as non negative matrix factorization, dictionary learning, compressive sensing with matrix uncertainty, robust principal component analysis, and sparse matrix factorization. Extensive numerical examples are provided to show that UAMPMF significantly outperforms state-of-the-art algorithms in terms of recovery accuracy, robustness and computational complexity.
We present automatic speech recognition (ASR) systems for Tamil and Kannada based on subword modeling to effectively handle unlimited vocabulary due to the highly agglutinative nature of the languages. We explore byte pair encoding (BPE), and proposed a variant of this algorithm named extended-BPE, and Morfessor tool to segment each word as subwords. We have effectively incorporated maximum likelihood (ML) and Viterbi estimation techniques with weighted finite state transducers (WFST) framework in these algorithms to learn the subword dictionary from a large text corpus. Using the learnt subword dictionary, the words in training data transcriptions are segmented to subwords and we train deep neural network ASR systems which recognize subword sequence for any given test speech utterance. The output subword sequence is then post-processed using deterministic rules to get the final word sequence such that the actual number of words that can be recognized is much larger. For Tamil ASR, We use 152 hours of data for training and 65 hours for testing, whereas for Kannada ASR, we use 275 hours for training and 72 hours for testing. Upon experimenting with different combination of segmentation and estimation techniques, we find that the word error rate (WER) reduces drastically when compared to the baseline word-level ASR, achieving a maximum absolute WER reduction of 6.24% and 6.63% for Tamil and Kannada respectively.
We present a method for supervised learning of sparsity-promoting regularizers for denoising signals and images. Sparsity-promoting regularization is a key ingredient in solving modern signal reconstruction problems; however, the operators underlying these regularizers are usually either designed by hand or learned from data in an unsupervised way. The recent success of supervised learning (mainly convolutional neural networks) in solving image reconstruction problems suggests that it could be a fruitful approach to designing regularizers. Towards this end, we propose to denoise signals using a variational formulation with a parametric, sparsity-promoting regularizer, where the parameters of the regularizer are learned to minimize the mean squared error of reconstructions on a training set of ground truth image and measurement pairs. Training involves solving a challenging bilievel optimization problem; we derive an expression for the gradient of the training loss using the closed-form solution of the denoising problem and provide an accompanying gradient descent algorithm to minimize it. Our experiments with structured 1D signals and natural images show that the proposed method can learn an operator that outperforms well-known regularizers (total variation, DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering for denoising. While the approach we present is specific to denoising, we believe that it could be adapted to the larger class of inverse problems with linear measurement models, giving it applicability in a wide range of signal reconstruction settings.