This paper addresses the fundamental question of when convex sets remain disjoint after random projection. We provide an analysis using ideas from high-dimensional convex geometry. For ellipsoids, we provide a bound in terms of the distance between these ellipsoids and simple functions of their polynomial coefficients. As an application, this theorem provides bounds for compressive classification of convex sets. Rather than assuming that the data to be classified is sparse, our results show that the data can be acquired via very few measurements yet will remain linearly separable. We demonstrate the feasibility of this approach in the context of hyperspectral imaging.
It is striking that, to date, all of the work in compressive classification has focused on classes of low dimension. This is perhaps an artifact of the mindset of compressed sensing, in which the projection preserves all information on coordinate planes of sufficiently small dimension. However, classification should not require nearly as much information as signal reconstruction does, and so we expect to be able to compressively classify into classes of full dimension; indeed, we allow two points in a common class to be mapped to the same compressive measurement, as this will not affect the classification. A Boolean version of this idea is studied in @cite_15 , which considers both random and optimality constructed projections. In the continuous setting, the closest existing work is that of Dasgupta @cite_12 @cite_7 , which uses random projections to learn a mixture of Gaussians. In particular, Dasgupta shows that sufficiently separated Gaussians stay separated after random projection. In the next section, we prove a similar result about ellipsoids, but with a sharper notion of separation.
@cite_15 @cite_7 @cite_12
2301486018 1531259569 1956647075
We compare and contrast from a geometric perspective a number of low-dimensional signal models that support stable information-preserving dimensionality reduction. We consider sparse and compressible signal models for deterministic and random signals, structured sparse and compressible signal models, point clouds, and manifold signal models. Each model has a particular geometrical structure that enables signal information to be stably preserved via a simple linear and nonadaptive projection to a much lower dimensional space; in each case the projection dimension is independent of the signal's ambient dimension at best or grows logarithmically with it at worst. As a bonus, we point out a common misconception related to probabilistic compressible signal models, namely, by showing that the oft-used generalized Gaussian and Laplacian models do not support stable linear dimensionality reduction. Compressive sampling (CS), also called compressed sensing, entails making observations of an unknown signal by projecting it onto random vectors. Recent theoretical results show that if the signal is sparse (or nearly sparse) in some basis, then with high probability such observations essentially encode the salient information in the signal. Further, the signal can be reconstructed from these "random projections," even when the number of observations is far less than the ambient signal dimension. The provable success of CS for signal reconstruction motivates the study of its potential in other applications. This paper investigates the utility of CS projection observations for signal classification (more specifically, m-ary hypothesis testing). Theoretical error bounds are derived and verified with several simulations. One main task for domain experts in analysing their nD data is to detect and interpret class cluster separations and outliers. In fact, an important question is, which features dimensions separate classes best or allow a cluster-based data classification. Common approaches rely on projections from nD to 2D, which comes with some challenges, such as: The space of projection contains an infinite number of items. How to find the right one? The projection approaches suffers from distortions and misleading effects. How to rely to the projected class cluster separation? The projections involve the complete set of dimensions features. How to identify irrelevant dimensions? Thus, to address these challenges, we introduce a visual analytics concept for the feature selection based on linear discriminative star coordinates DSC, which generate optimal cluster separating views in a linear sense for both labeled and unlabeled data. This way the user is able to explore how each dimension contributes to clustering. To support to explore relations between clusters and data dimensions, we provide a set of cluster-aware interactions allowing to smartly iterate through subspaces of both records and features in a guided manner. We demonstrate our features selection approach for optimal cluster class separation analysis with a couple of experiments on real-life benchmark high-dimensional data sets. In passive monitoring using sensor networks, low energy supplies drastically constrain sensors in terms of calculation and communication abilities. Designing processing algorithms at the sensor level that take into account these constraints is an important problem in this context. Here we study the estimation of correlation functions between sensors using compressed acquisition and one-bit-quantization. The estimation is achieved directly using compressed samples, without considering any reconstruction of the signals. We show that if the signals of interest are far from white noise, estimation of the correlation using @math compressed samples out of @math can be more advantageous than estimation of the correlation using @math consecutive samples. The analysis consists of studying the asymptotic performance of the estimators at a fixed compression rate. We provide the analysis when the compression is realized by a random projection matrix composed of independent and identically distributed entries. The framework includes widely used random projection matrices, such as Gaussian and Bernoulli matrices, and it also includes very sparse matrices. However, it does not include subsampling without replacement, for which a separate analysis is provided. When considering one-bit-quantization as well, the theoretical analysis is not tractable. However, empirical evidence allows the conclusion that in practical situations, compressed and quantized estimators behave sufficiently correctly to be useful in, for example, time-delay estimation and model estimation. The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement scheme. This has inspired the design of hardware systems that directly implement random measurement protocols. However, despite the intense focus of the community on signal recovery, many (if not most) signal processing problems do not require full signal recovery. In this paper, we take some first steps in the direction of solving inference problems-such as detection, classification, or estimation-and filtering problems using only compressive measurements and without ever reconstructing the signals involved. We provide theoretical bounds along with experimental results. The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible image or signal from a small set of linear, non-adaptive (even random) projections. However, in many applications, including object and target recognition, we are ultimately interested in making a decision about an image rather than computing a reconstruction. We propose here a framework for compressive classification that operates directly on the compressive measurements without first reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the compressive domain; we find that the number of measurements required for a given classification performance level does not depend on the sparsity or compressibility of the images but only on the noise level. The second part of the theory applies the generalized maximum likelihood method to deal with unknown transformations such as the translation, scale, or viewing angle of a target object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear manifold in the high-dimensional image space. We find that the number of measurements required for a given classification performance level grows linearly in the dimensionality of the manifold but only logarithmically in the number of pixels samples and image classes. Using both simulations and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness of the smashed filter for target classification using very few measurements. Compressed sensing (CS) is a sampling theory that allows reconstruction of sparse (or compressible) signals from an incomplete number of measurements, using of a sensing mechanism implemented by an appropriate projection matrix. The CS theory is based on random Gaussian projection matrices, which satisfy recovery guarantees with high probability; however, sparse ternary 0, -1, +1 projections are more suitable for hardware implementation. In this paper, we present a deep learning approach to obtain very sparse ternary projections for compressed sensing. Our deep learning architecture jointly learns a pair of a projection matrix and a reconstruction operator in an end-to-end fashion. The experimental results on real images demonstrate the effectiveness of the proposed approach compared to state-of-the-art methods, with significant advantage in terms of complexity. We introduce a new signal model, called (K,C)-sparse, to capture K-sparse signals in N dimensions whose nonzero coefficients are contained within at most C clusters, with C < K << N. In contrast to the existing work in the sparse approximation and compressive sensing literature on block sparsity, no prior knowledge of the locations and sizes of the clusters is assumed. We prove that O (K + C log(N C))) random projections are sufficient for (K,C)-model sparse signal recovery based on subspace enumeration. We also provide a robust polynomialtime recovery algorithm for (K,C)-model sparse signals with provable estimation guarantees. The recently introduced theory of compressed sensing (CS) enables the reconstruction or approximation of sparse or compressible signals from a small set of incoherent projections; often the number of projections can be much smaller than the number of Nyquist rate samples. In this paper, we show that the CS framework is information scalable to a wide range of statistical inference tasks. In particular, we demonstrate how CS principles can solve signal detection problems given incoherent measurements without ever reconstructing the signals involved. We specifically study the case of signal detection in strong inference and noise and propose an incoherent detection and estimation algorithm (IDEA) based on matching pursuit. The number of measurements and computations necessary for successful detection using IDEA is significantly lower than that necessary for successful reconstruction. Simulations show that IDEA is very resilient to strong interference, additive noise, and measurement quantization. When combined with random measurements, IDEA is applicable to a wide range of different signal classes Compressed sensing (CS) offers a joint compression and sensing processes, based on the existence of a sparse representation of the treated signal and a set of projected measurements. Work on CS thus far typically assumes that the projections are drawn at random. In this paper, we consider the optimization of these projections. Since such a direct optimization is prohibitive, we target an average measure of the mutual coherence of the effective dictionary, and demonstrate that this leads to better CS reconstruction performance. Both the basis pursuit (BP) and the orthogonal matching pursuit (OMP) are shown to benefit from the newly designed projections, with a reduction of the error rate by a factor of 10 and beyond. We relate compressed sensing (CS) with Bayesian experimental design and provide a novel efficient approximate method for the latter, based on expectation propagation. In a large comparative study about linearly measuring natural images, we show that the simple standard heuristic of measuring wavelet coefficients top-down systematically outperforms CS methods using random measurements; the sequential projection optimisation approach of (Ji & Carin, 2007) performs even worse. We also show that our own approximate Bayesian method is able to learn measurement filters on full images efficiently which outperform the wavelet heuristic. To our knowledge, ours is the first successful attempt at "learning compressed sensing" for images of realistic size. In contrast to common CS methods, our framework is not restricted to sparse signals, but can readily be applied to other notions of signal complexity or noise models. We give concrete ideas how our method can be scaled up to large signal representations. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala [30], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Renyi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For “discrete” signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal pX. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal @math . We study the notion of compressed sensing (CS) as put forward by Donoho, Candes, Tao and others. The notion proposes a signal or image, unknown but supposed to be compressible by a known transform, (e.g. wavelet or Fourier), can be subjected to fewer measurements than the nominal number of data points, and yet be accurately reconstructed. The samples are nonadaptive and measure 'random' linear combinations of the transform coefficients. Approximate reconstruction is obtained by solving for the transform coefficients consistent with measured data and having the smallest possible l1 norm.We present initial 'proof-of-concept' examples in the favorable case where the vast majority of the transform coefficients are zero. We continue with a series of numerical experiments, for the setting of lp-sparsity, in which the object has all coefficients nonzero, but the coefficients obey an lp bound, for some p ∈ (0, 1]. The reconstruction errors obey the inequalities paralleling the theory, seemingly with well-behaved constants.We report that several workable families of 'random' linear combinations all behave equivalently, including random spherical, random signs, partial Fourier and partial Hadamard.We next consider how these ideas can be used to model problems in spectroscopy and image processing, and in synthetic examples see that the reconstructions from CS are often visually "noisy". To suppress this noise we postprocess using translation-invariant denoising, and find the visual appearance considerably improved.We also consider a multiscale deployment of compressed sensing, in which various scales are segregated and CS applied separately to each; this gives much better quality reconstructions than a literal deployment of the CS methodology.These results show that, when appropriately deployed in a favorable setting, the CS framework is able to save significantly over traditional sampling, and there are many useful extensions of the basic idea. In sparse approximation theory, the fundamental problem is to reconstruct a signal A∈ℝn from linear measurements 〈Aψi〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1 e, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + e and improves the reconstruction time from poly(n) to poly(k logn) Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by 1-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by 1-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a log-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality. In recent years analysis of complexity of learning Gaussian mixture models from sampled data has received significant attention in computational machine learning and theory communities. In this paper we present the first result showing that polynomial time learning of multidimensional Gaussian Mixture distributions is possible when the separation between the component means is arbitrarily small. Specifically, we present an algorithm for learning the parameters of a mixture of k identical spherical Gaussians in n-dimensional space with an arbitrarily small separation between the components, which is polynomial in dimension, inverse component separation and other input parameters for a fixed number of components k. The algorithm uses a projection to k dimensions and then a reduction to the 1-dimensional case. It relies on a theoretical analysis showing that two 1-dimensional mixtures whose densities are close in the L norm must have similar means and mixing coefficients. To produce the necessary lower bound for the L norm in terms of the distances between the corresponding means, we analyze the behavior of the Fourier transform of a mixture of Gaussians in one dimension around the origin, which turns out to be closely related to the properties of the Vandermonde matrix obtained from the component means. Analysis of minors of the Vandermonde matrix together with basic function approximation results allows us to provide a lower bound for the norm of the mixture in the Fourier domain and hence a bound in the original space. Additionally, we present a separate argument for reconstructing variance. Three important properties of a classification machinery are i) the system preserves the core information of the input data; ii) the training examples convey information about unseen data; and iii) the system is able to treat differently points from different classes. In this paper, we show that these fundamental properties are satisfied by the architecture of deep neural networks. We formally prove that these networks with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data. Similar points at the input of the network are likely to have a similar output. The theoretical analysis of deep networks here presented exploits tools used in the compressed sensing and dictionary learning literature, thereby making a formal connection between these important topics. The derived results allow drawing conclusions on the metric learning properties of the network and their relation to its structure, as well as providing bounds on the required size of the training set such that the training examples would represent faithfully the unseen data. The results are validated with state-of-the-art trained networks. Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a sparse version of the fundamental tool in dimension reduction -- the Johnson-Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just O(1 e) non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of Ω(1 e2) both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an O(1 e) update time per non-zero element for a (1 e)-approximate projection, thereby substantially outperforming the O(1 e2) update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its O(d) worst case running time matches the best approach of Ailon and Liberty. This brief paper presents a novel method for low-resolution face recognition. We introduce a generalized bipartite graph to discretely approximate the underlying manifold structure of face sets with different resolutions. Unlike traditional graph-based methods that only construct the graph based on one sample set, the proposed method constructs the generalized bipartite graph on two heterogeneous sample sets and contains more completed information. Our method learns a couple of mappings that project the face sets with different dimensions into a unified feature space which favors the task of classification. Specifically, in this unified space, our method preserves within-class local geometrical structure according to the network topology of the generalized bipartite graph and maximizes between-class separability at the same time. Experimental results on two benchmark face databases demonstrate the effectiveness of our proposed algorithm. HighlightsWe propose a novel method for low-resolution face recognition.We introduce the generalized bipartite graph to approximate the manifold structure.Our method preserves within-class local geometrical structure.Our method maximizes between-class separability.Low and high resolution faces are projected to a unified discriminative subspace.