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
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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. 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 Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex optimization problems. In this paper, we propose the use of the alternating direction method - a classic approach for optimization problems with separable variables (D. Gabay and B. Mercier, ?A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,? Computer and Mathematics with Applications, vol. 2, pp. 17-40, 1976; R. Glowinski and A. Marrocco, ?Sur lapproximation par elements finis dordre un, et la resolution par penalisation-dualite dune classe de problemes de Dirichlet nonlineaires,? Rev. Francaise dAut. Inf. Rech. Oper., vol. R-2, pp. 41-76, 1975) - for signal reconstruction from partial Fourier (i.e., incomplete frequency) measurements. Signals are reconstructed as minimizers of the sum of three terms corresponding to total variation, ?1-norm of a certain transform, and least squares data fitting. Our algorithm, called RecPF and published online, runs very fast (typically in a few seconds on a laptop) because it requires a small number of iterations, each involving simple shrinkages and two fast Fourier transforms (or alternatively discrete cosine transforms when measurements are in the corresponding domain). RecPF was compared with two state-of-the-art algorithms on recovering magnetic resonance images, and the results show that it is highly efficient, stable, and robust. In this paper, we propose a new classification method for five categories of lung tissues in high-resolution computed tomography (HRCT) images, with feature-based image patch approximation. We design two new feature descriptors for higher feature descriptiveness, namely the rotation-invariant Gabor-local binary patterns (RGLBP) texture descriptor and multi-coordinate histogram of oriented gradients (MCHOG) gradient descriptor. Together with intensity features, each image patch is then labeled based on its feature approximation from reference image patches. And a new patch-adaptive sparse approximation (PASA) method is designed with the following main components: minimum discrepancy criteria for sparse-based classification, patch-specific adaptation for discriminative approximation, and feature-space weighting for distance computation. The patch-wise labelings are then accumulated as probabilistic estimations for region-level classification. The proposed method is evaluated on a publicly available ILD database, showing encouraging performance improvements over the state-of-the-arts. Abstract It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This technique known as “compressed sensing” or “compressive sampling” relies on properties of the sensing matrix such as the restricted isometry property . In this Note, we establish new results about the accuracy of the reconstruction from undersampled measurements which improve on earlier estimates, and have the advantage of being more elegant. To cite this article: E.J. Candes, C. R. Acad. Sci. Paris, Ser. I 346 (2008). This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all standard models-e.g., Gaussian, frequency measurements-discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) to hold near the sparsity level in question, nor a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s logn Fourier coefficients that are contaminated with noise. This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements. 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 . The goal of sparse recovery is to recover a k-sparse signal x e Rn from (possibly noisy) linear measurements of the form y = Ax, where A e Rmxn describes the measurement process. Standard results in compressive sensing show that it is possible to recover the signal x from m = O(k log(n k)) measurements, and that this bound is tight. The framework of model-based compressive sensing [BCDH10] overcomes the lower bound and reduces the number of measurements further to O(k) by limiting the supports of x to a subset M of the (nk) possible supports. This has led to many measurement-efficient algorithms for a wide variety of signal models, including block-sparsity and tree-sparsity. Unfortunately, extending the framework to other, more general models has been stymied by the following obstacle: for the framework to apply, one needs an algorithm that, given a signal x, solves the following optimization problem exactly: [EQUATION] (here x[n] denotes the projection of x on coordinates not in Ω). However, an approximation algorithm for this optimization task is not sufficient. Since many problems of this form are not known to have exact polynomial-time algorithms, this requirement poses an obstacle for extending the framework to a richer class of models. In this paper, we remove this obstacle and show how to extend the model-based compressive sensing framework so that it requires only approximate solutions to the aforementioned optimization problems. Interestingly, our extension requires the existence of approximation algorithms for both the maximization and the minimization variants of the optimization problem. Further, we apply our framework to the Constrained Earth Mover's Distance (CEMD) model introduced in [SHI13], obtaining a sparse recovery scheme that uses significantly less than O(k log(n k)) measurements. This is the first non-trivial theoretical bound for this model, since the validation of the approach presented in [SHI13] was purely empirical. The result is obtained by designing a novel approximation algorithm for the maximization version of the problem and proving approximation guarantees for the minimization algorithm described in [SHI13]. The theory of compressive sensing (CS) has shown us that under certain conditions, a sparse signal can be recovered from a small number of linear incoherent measurements. An effective class of reconstruction algorithms involve solving a convex optimization program that balances the l1 norm of the solution against a data fidelity term. Tremendous progress has been made in recent years on algorithms for solving these l1 minimization programs. These algorithms, however, are for the most part static: they focus on finding the solution for a fixed set of measurements. In this paper, we present a suite of dynamic algorithms for solving l1 minimization programs for streaming sets of measurements. We consider cases where the underlying signal changes slightly between measurements, and where new measurements of a fixed signal are sequentially added to the system. We develop algorithms to quickly update the solution of several different types of l1 optimization problems whenever these changes occur, thus avoiding having to solve a new optimization problem from scratch. Our proposed schemes are based on homotopy continuation, which breaks down the solution update in a systematic and efficient way into a small number of linear steps. Each step consists of a low-rank update and a small number of matrix-vector multiplications - very much like recursive least squares. Our investigation also includes dynamic updating schemes for l1 decoding problems, where an arbitrary signal is to be recovered from redundant coded measurements which have been corrupted by sparse errors. 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. 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. Fine-grained visual categorization (FGVC) is to categorize objects into subordinate classes instead of basic classes. One major challenge in FGVC is the co-occurrence of two issues: 1) many subordinate classes are highly correlated and are difficult to distinguish, and 2) there exists the large intra-class variation (e.g., due to object pose). This paper proposes to explicitly address the above two issues via distance metric learning (DML). DML addresses the first issue by learning an embedding so that data points from the same class will be pulled together while those from different classes should be pushed apart from each other; and it addresses the second issue by allowing the flexibility that only a portion of the neighbors (not all data points) from the same class need to be pulled together. However, feature representation of an image is often high dimensional, and DML is known to have difficulty in dealing with high dimensional feature vectors since it would require @math for storage and @math for optimization. To this end, we proposed a multi-stage metric learning framework that divides the large-scale high dimensional learning problem to a series of simple subproblems, achieving @math computational complexity. The empirical study with FVGC benchmark datasets verifies that our method is both effective and efficient compared to the state-of-the-art FGVC approaches. 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. 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. Traditional classification systems rely heavily on sufficient training data with accurate labels. However, the quality of the collected data depends on the labelers, among which inexperienced labelers may exist and produce unexpected labels that may degrade the performance of a learning system. In this paper, we investigate the multiclass classification problem where a certain amount of training examples are randomly labeled. Specifically, we show that this issue can be formulated as a label noise problem. To perform multiclass classification, we employ the widely used importance reweighting strategy to enable the learning on noisy data to more closely reflect the results on noise-free data. We illustrate the applicability of this strategy to any surrogate loss functions and to different classification settings. The proportion of randomly labeled examples is proved to be upper bounded and can be estimated under a mild condition. The convergence analysis ensures the consistency of the learned classifier to the optimal classifier with respect to clean data. Two instantiations of the proposed strategy are also introduced. Experiments on synthetic and real data verify that our approach yields improvements over the traditional classifiers as well as the robust classifiers. Moreover, we empirically demonstrate that the proposed strategy is effective even on asymmetrically noisy data. Compressive Sensing theory details how a sparsely represented signal in a known basis can be reconstructed with an underdetermined linear measurement model. However, in reality there is a mismatch between the assumed and the actual bases due to factors such as discretization of the parameter space defining basis components, sampling jitter in A D conversion, and model errors. Due to this mismatch, a signal may not be sparse in the assumed basis, which causes significant performance degradation in sparse reconstruction algorithms. To eliminate the mismatch problem, this paper presents a novel perturbed orthogonal matching pursuit (POMP) algorithm that performs controlled perturbation of selected support vectors to decrease the orthogonal residual at each iteration. Based on detailed mathematical analysis, conditions for successful reconstruction are derived. Simulations show that robust results with much smaller reconstruction errors in the case of perturbed bases can be obtained as compared to standard sparse reconstruction techniques. Breakthrough results in compressive sensing (CS) have shown that high dimensional signals (vectors) can often be accurately recovered from a relatively small number of non-adaptive linear projection observations, provided that they possess a sparse representation in some basis. Subsequent efforts have established that the reconstruction performance of CS can be improved by employing additional prior signal knowledge, such as dependency in the location of the non-zero signal coefficients (structured sparsity) or by collecting measurements sequentially and adaptively, in order to focus measurements into the proper subspace where the unknown signal resides. In this paper, we examine a powerful hybrid of adaptivity and structure. We identify a particular form of structured sparsity that is amenable to adaptive sensing, and using concepts from sparse hierarchical dictionary learning we demonstrate that sparsifying dictionaries exhibiting the appropriate form of structured sparsity can be learned from a collection of training data. The combination of these techniques (structured dictionary learning and adaptive sensing) results in an effective and efficient adaptive compressive acquisition approach which we refer to as LASeR (Learning Adaptive Sensing Representations.) In this paper, we focus on improving the proposal classification stage in the object detection task and present implicit negative sub-categorization and sink diversion to lift the performance by strengthening loss function in this stage. First, based on the observation that the “background” class is generally very diverse and thus challenging to be handled as a single indiscriminative class in existing state-of-the-art methods, we propose to divide the background category into multiple implicit sub-categories to explicitly differentiate diverse patterns within it. Second, since the ground truth class inevitably has low-value probability scores for certain images, we propose to add a “sink” class and divert the probabilities of wrong classes to this class when necessary, such that the ground truth label will still have a higher probability than other wrong classes even though it has low probability output. Additionally, we propose to use dilated convolution, which is widely used in the semantic segmentation task, for efficient and valuable context information extraction. Extensive experiments on PASCAL VOC 2007 and 2012 data sets show that our proposed methods based on faster R-CNN implementation can achieve state-of-the-art mAPs, i.e., 84.1 , 82.6 , respectively, and obtain 2.5 improvement on ILSVRC DET compared with that of ResNet. Inspired by theories of sparse representation and compressed sensing, this paper presents a simple, novel, yet very powerful approach for texture classification based on random projection, suitable for large texture database applications. At the feature extraction stage, a small set of random features is extracted from local image patches. The random features are embedded into a bag--of-words model to perform texture classification; thus, learning and classification are carried out in a compressed domain. The proposed unconventional random feature extraction is simple, yet by leveraging the sparse nature of texture images, our approach outperforms traditional feature extraction methods which involve careful design and complex steps. We have conducted extensive experiments on each of the CUReT, the Brodatz, and the MSRC databases, comparing the proposed approach to four state-of-the-art texture classification methods: Patch, Patch-MRF, MR8, and LBP. We show that our approach leads to significant improvements in classification accuracy and reductions in feature dimensionality.