A learning-based framework for representation of domain-specific images is proposed where joint compression and denoising can be done using a VQ-based multi-layer network. While it learns to compress the images from a training set, the compression performance is very well generalized on images from a test set. Moreover, when fed with noisy versions of the test set, since it has priors from clean images, the network also efficiently denoises the test images during the reconstruction. The proposed framework is a regularized version of the Residual Quantization (RQ) where at each stage, the quantization error from the previous stage is further quantized. Instead of codebook learning from the k-means which over-trains for high-dimensional vectors, we show that only generating the codewords from a random, but properly regularized distribution suffices to compress the images globally and without the need to resort to patch-based division of images. The experiments are done on the set of facial images and the method is compared with the JPEG-2000 codec for compression and BM3D for denoising, showing promising results.
Depending on @math and @math , the problem of Eq. can be treated in many different ways. See @cite_11 and @cite_9 for detailed reviews and discussions For example, under the famous sparsity constraint @math or its relaxed version @math , the K-SVD algorithm @cite_7 solves it for local minima in an iterative way.
@cite_9 @cite_7 @cite_11
2962984690 2318684976 2766596763 1554201860
We consider processing an n x d matrix A in a stream with row-wise updates according to a recent algorithm called Frequent Directions (Liberty, KDD 2013). This algorithm maintains an e x d matrix Q deterministically, processing each row in O(de2) time; the processing time can be decreased to O(de) with a slight modification in the algorithm and a constant increase in space. Then for any unit vector x, the matrix Q satisfies [EQUATION] We show that if one sets e = ⌈k + k e⌉ and returns Qk, a k x d matrix that is simply the top k rows of Q, then we achieve the following properties: [EQUATION] and where πQk (A) is the projection of A onto the rowspace of Qk then [EQUATION] We also show that Frequent Directions cannot be adapted to a sparse version in an obvious way that retains e original rows of the matrix, as opposed to a linear combination or sketch of the rows. In this article we present the first effective method based on global optimization for the reconstruction of image puzzles comprising rectangle pieces—Puzzle Solving by Quadratic Programming (PSQP). The proposed novel mathematical formulation reduces the problem to the maximization of a constrained quadratic function, which is solved via a gradient ascent approach. The proposed method is deterministic and can deal with arbitrary identical rectangular pieces. We provide experimental results showing its effectiveness when compared to state-of-the-art approaches. Although the method was developed to solve image puzzles, we also show how to apply it to the reconstruction of simulated strip-shredded documents, broadening its applicability. The recent years have witnessed the emerging of vector quantization (VQ) techniques for efficient similarity search. VQ partitions the feature space into a set of codewords and encodes data points as integer indices using the codewords. Then the distance between data points can be efficiently approximated by simple memory lookup operations. By the compact quantization, the storage cost, and searching complexity are significantly reduced, thereby facilitating efficient large-scale similarity search. However, the performance of several celebrated VQ approaches degrades significantly when dealing with noisy data. In addition, it can barely facilitate a wide range of applications as the distortion measurement only limits to @math norm . To address the shortcomings of the squared Euclidean ( @math norm) loss function employed by the VQ approaches, in this paper, we propose a novel robust and general VQ framework, named RGVQ, to enhance both robustness and generalization of VQ approaches. Specifically, a @math -norm loss function is proposed to conduct the @math -norm similarity search, rather than the @math norm search, and the @math -th order loss is used to enhance the robustness. Despite the fact that changing the loss function to @math norm makes VQ approaches more robust and generic, it brings us a challenge that a non-smooth and non-convex orthogonality constrained @math - norm function has to be minimized. To solve this problem, we propose a novel and efficient optimization scheme and specify it to VQ approaches and theoretically prove its convergence. Extensive experiments on benchmark data sets demonstrate that the proposed RGVQ is better than the original VQ for several approaches, especially when searching similarity in noisy data. Given a matrix A e ℝ m ×n of rank r, and an integer k < r, the top k singular vectors provide the best rank-k approximation to A. When the columns of A have specific meaning, it is desirable to find (provably) "good" approximations to A k which use only a small number of columns in A. Proposed solutions to this problem have thus far focused on randomized algorithms. Our main result is a simple greedy deterministic algorithm with guarantees on the performance and the number of columns chosen. Specifically, our greedy algorithm chooses c columns from A with @math such that @math where C gr is the matrix composed of the c columns, @math is the pseudo-inverse of C gr ( @math is the best reconstruction of A from C gr), and ¼(A) is a measure of the coherence in the normalized columns of A. The running time of the algorithm is O(SVD(A k) + mnc) where SVD(A k) is the running time complexity of computing the first k singular vectors of A. To the best of our knowledge, this is the first deterministic algorithm with performance guarantees on the number of columns and a (1 + e) approximation ratio in Frobenius norm. The algorithm is quite simple and intuitive and is obtained by combining a generalization of the well known sparse approximation problem from information theory with an existence result on the possibility of sparse approximation. Tightening the analysis along either of these two dimensions would yield improved results.