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Register a new userShort-and-Sparse Deconvolution -- A Geometric Approach
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Short-and-sparse deconvolution (SaSD) is the problem of extracting localized, recurring motifs in signals with spatial or temporal structure. Variants of this problem arise in applications such as image deblurring, microscopy, neural spike sorting, and more. The problem is challenging in both theory and practice, as natural optimization formulations are nonconvex. Moreover, practical deconvolution problems involve smooth motifs (kernels) whose spectra decay rapidly, resulting in poor conditioning and numerical challenges. This paper is motivated by recent theoretical advances, which characterize the optimization landscape of a particular nonconvex formulation of SaSD. This is used to derive a $provable$ algorithm which exactly solves certain non-practical instances of the SaSD problem. We leverage the key ideas from this theory (sphere constraints, data-driven initialization) to develop a $practical$ algorithm, which performs well on data arising from a range of application areas. We highlight key additional challenges posed by the ill-conditioning of real SaSD problems, and suggest heuristics (acceleration, continuation, reweighting) to mitigate them. Experiments demonstrate both the performance and generality of the proposed method.
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In this paper, we incorporate a graph filter deconvolution step into the classical geometric convolutional neural network pipeline. More precisely, under the assumption that the graph domain plays a role in the generation of the observed graph signals, we pre-process every signal by passing it through a sparse deconvolution operation governed by a pre-specified filter bank. This deconvolution operation is formulated as a group-sparse recovery problem, and convex relaxations that can be solved efficiently are put forth. The deconvolved signals are then fed into the geometric convolutional neural network, yielding better classification performance than their unprocessed counterparts. Numerical experiments showcase the effectiveness of the deconvolution step on classification tasks on both synthetic and real-world settings.
We propose a blind deconvolution method for signals on graphs, with the exact sparseness constraint for the original signal. Graph blind deconvolution is an algorithm for estimating the original signal on a graph from a set of blurred and noisy measurements. Imposing a constraint on the number of nonzero elements is desirable for many different applications. This paper deals with the problem with constraints placed on the exact number of original sources, which is given by an optimization problem with an $ell_0$ norm constraint. We solve this non-convex optimization problem using the ADMM iterative solver. Numerical experiments using synthetic signals demonstrate the effectiveness of the proposed method.
In the blind deconvolution problem, we observe the convolution of an unknown filter and unknown signal and attempt to reconstruct the filter and signal. The problem seems impossible in general, since there are seemingly many more unknowns than knowns . Nevertheless, this problem arises in many application fields; and empirically, some of these fields have had success using heuristic methods -- even economically very important ones, in wireless communications and oil exploration. Todays fashionable heuristic formulations pose non-convex optimization problems which are then attacked heuristically as well. The fact that blind deconvolution can be solved under some repeatable and naturally-occurring circumstances poses a theoretical puzzle. To bridge the gulf between reported successes and theorys limited understanding, we exhibit a convex optimization problem that -- assuming signal sparsity -- can convert a crude approximation to the true filter into a high-accuracy recovery of the true filter. Our proposed formulation is based on L1 minimization of inverse filter outputs. We give sharp guarantees on performance of the minimizer assuming sparsity of signal, showing that our proposal precisely recovers the true inverse filter, up to shift and rescaling. There is a sparsity/initial accuracy tradeoff: the less accurate the initial approximation, the greater we rely on sparsity to enable exact recovery. To our knowledge this is the first reported tradeoff of this kind. We consider it surprising that this tradeoff is independent of dimension. We also develop finite-$N$ guarantees, for highly accurate reconstruction under $Ngeq O(k log(k) )$ with high probability. We further show stable approximation when the true inverse filter is infinitely long and extend our guarantees to the case where the observations are contaminated by stochastic or adversarial noise.
To better retain the deep features of an image and solve the sparsity problem of the end-to-end segmentation model, we propose a new deep convolutional network model for medical image pixel segmentation, called MC-Net. The core of this network model consists of four parts, namely, an encoder network, a multiple max-pooling integration module, a cross multiscale deconvolution decoder network and a pixel-level classification layer. In the network structure of the encoder, we use multiscale convolution instead of the traditional single-channel convolution. The multiple max-pooling integration module first integrates the output features of each submodule of the encoder network and reduces the number of parameters by convolution using a kernel size of 1. At the same time, each max-pooling layer (the pooling size of each layer is different) is spliced after each convolution to achieve the translation invariance of the feature maps of each submodule. We use the output feature maps from the multiple max-pooling integration module as the input of the decoder network; the multiscale convolution of each submodule in the decoder network is cross-fused with the feature maps generated by the corresponding multiscale convolution in the encoder network. Using the above feature map processing methods solves the sparsity problem after the max-pooling layer-generating matrix and enhances the robustness of the classification. We compare our proposed model with the well-known Fully Convolutional Networks for Semantic Segmentation (FCNs), DecovNet, PSPNet, U-net, SgeNet and other state-of-the-art segmentation networks such as HyperDenseNet, MS-Dual, Espnetv2, Denseaspp using one binary Kaggle 2018 data science bowl dataset and two multiclass dataset and obtain encouraging experimental results.
ProSper is a python library containing probabilistic algorithms to learn dictionaries. Given a set of data points, the implemented algorithms seek to learn the elementary components that have generated the data. The library widens the scope of dictionary learning approaches beyond implementations of standard approaches such as ICA, NMF or standard L1 sparse coding. The implemented algorithms are especially well-suited in cases when data consist of components that combine non-linearly and/or for data requiring flexible prior distributions. Furthermore, the implemented algorithms go beyond standard approaches by inferring prior and noise parameters of the data, and they provide rich a-posteriori approximations for inference. The library is designed to be extendable and it currently includes: Binary Sparse Coding (BSC), Ternary Sparse Coding (TSC), Discrete Sparse Coding (DSC), Maximal Causes Analysis (MCA), Maximum Magnitude Causes Analysis (MMCA), and Gaussian Sparse Coding (GSC, a recent spike-and-slab sparse coding approach). The algorithms are scalable due to a combination of variational approximations and parallelization. Implementations of all algorithms allow for parallel execution on multiple CPUs and multiple machines for medium to large-scale applications. Typical large-scale runs of the algorithms can use hundreds of CPUs to learn hundreds of dictionary elements from data with tens of millions of floating-point numbers such that models with several hundred thousand parameters can be optimized. The library is designed to have minimal dependencies and to be easy to use. It targets users of dictionary learning algorithms and Machine Learning researchers.
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