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Data compression techniques are characterized by four key performance indices which are (i) associated accuracy, (ii) compression ratio, (iii) computational work, and (iv) degree of freedom. The method of data compression developed in this paper allows us to substantially improve all the four issues above. The proposed transform $F$ is presented in the form of a sum with $p-1$ terms, $F_1,..., F_{p-1}$, where each term is a particular sub-transform presented by a first degree polynomial. For $j=1,\ldots,p-1$, each sub-transform $F_{j}$ is determined from interpolation-like conditions. This device provides the transform flexibility to incorporate variation of observed data and leads to performance improvement. The transform $F$ has two degrees of freedom, the number of sub-transforms and associated compression ratio associated with each sub-transform $F_{j}$.

Suppose $K_{_Y}$ and $K_{_X}$ are large sets of observed and reference signals, respectively, each containing $N$ signals. Is it possible to construct a filter $F$ that requires a priori information only on few signals, $p\ll N$, from $K_{_X}$ but performs better than the known filters based on a priori information on every reference signal from $K_{_X}$? It is shown that the positive answer is achievable under quite unrestrictive assumptions. The device behind the proposed method is based on a special extension of the piecewise linear interpolation technique to the case of random signal sets. The proposed technique provides a single filter to process any signal from the arbitrarily large signal set. The filter is determined in terms of pseudo-inverse matrices so that it always exists.

In terms of signal samples, we propose and justify a new rank reduced multi-term transform, abbreviated as MTT, which, under certain conditions, may provide better-associated accuracy than that of known optimal rank reduced transforms. The basic idea is to construct the transform with more parameters to optimize than those in the known optimal transforms. This is realized by the extension of the known transform structures to the form that includes additional terms - the MTT has four matrices to minimize the cost. The MTT structure has also a special transformation that decreases the numerical load. As a result, the MTT performance is improved by the variation of the MTT components.

We study the problem of multi-compression and reconstructing a stochastic signal observed by several independent sensors (or compressors) that transmit compressed information to a fusion center. { The key aspect of this problem is to find models of the sensors and fusion center that are optimized in the sense of an error minimization under a certain criterion, such as the mean square error (MSE).} { A novel technique to solve this problem is developed. The novelty is as follows. First, the multi-compressors are non-linear and modeled using second degree polynomials. This may increase the accuracy of the signal estimation through the optimization in a higher dimensional parameter space compared to the linear case. Second, the required models are determined by a method based on a combination of the second degree transform (SDT) with the maximum block improvement (MBI) method and the generalized rank-constrained matrix approximation. It allows us to use the advantages of known methods to further increase the estimation accuracy of the source signal. Third, the proposed method is justified in terms of pseudo-inverse matrices. As a result, the models of compressors and fusion center always exist and are numerically stable.} In other words, the proposed models may provide compression, de-noising and reconstruction of distributed signals in cases when known methods either are not applicable or may produce larger associated errors., Comment: arXiv admin note: text overlap with arXiv:1508.04514

Principal Component Analysis (PCA) is a transform for finding the principal components (PCs) that represent features of random data. PCA also provides a reconstruction of the PCs to the original data. We consider an extension of PCA which allows us to improve the associated accuracy and diminish the numerical load, in comparison with known techniques. This is achieved due to the special structure of the proposed transform which contains two matrices $T_0$ and $T_1$, and a special transformation $\mathcal{f}$ of the so called auxiliary random vector $\mathbf w$. For this reason, we call it the three-term PCA. In particular, we show that the three-term PCA always exists, i.e. is applicable to the case of singular data. Both rigorous theoretical justification of the three-term PCA and simulations with real-world data are provided.

We propose an approach to the estimation of infinite sets of random vectors. The problem addressed is as follows. Given two infinite sets of random vectors, find a single estimator that estimates vectors from with a controlled associated error. A new theory for the existence and implementation of such an estimator is studied. In particular, we show that the proposed estimator is asymptotically optimal. Moreover, the estimator is determined in terms of pseudo-inverse matrices and, therefore, it always exists.

While the theory of operator approximation with any given accuracy is well elaborated, the theory of {best constrained} constructive operator approximation is still not so well developed. Despite increasing demands from applications this subject is hardly tractable because of intrinsic difficulties in associated approximation techniques. This paper concerns the best constrained approximation of a non-linear operator in probability spaces. We propose and justify a new approach and technique based on the following observation. Methods for best approximation are aimed at obtaining the best solution within a certain class; the accuracy of the solution is limited by the extent to which the class is suitable. By contrast, iterative methods are normally convergent but the convergence can be quite slow. Moreover, in practice only a finite number of iteration loops can be carried out and therefore, the final approximate solution is often unsatisfactorily inaccurate. A natural idea is to combine the methods for best approximation and iterative techniques to exploit their advantageous features. Here, we present an approach which realizes this. The proposed approximating operator has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the approximating technique we develop is special random vectors called injections. They are determined in the way that allows us to further minimize the associated error., Comment: 27 pages

We consider the case when a set of spatially distributed sensors make local observations which are noisy versions of a signal of interest. Each sensor transmits compressed information about its measurements to the fusion center which should recover the original signal within a prescribed accuracy. Such an information processing relates to a wireless sensor network (WSN) scenario. The key problem is to find models of the sensors and fusion center so that they will be optimal in the sense of minimization of the associated error under a certain criterion, such as the mean square error (MSE). We determine the models from the technique which is a combination of the maximum block improvement (MBI) method and the generic Karhunen-Lo\`{e}ve transform (KLT). The resulting multi-compressor KLT-MBI algorithm is given in terms of pseudo-inverse matrices and, therefore, it is numerically stable and always exists. The proposed model provides compression, de-noising and reconstruction of distributed signals for the cases when known methods either are not applicable (because of singularity of associated matrices) or produce larger associated errors. Error analysis is provided.

We propose and justify a new approach to constructing optimal nonlinear transforms of random vectors. We show that the proposed transform improves such characteristics of rank-reduced transforms as compression ratio, accuracy of decompression and reduces required computational work. The proposed transform ${\mathcal T}_p$ is presented in the form of a sum with $p$ terms where each term is interpreted as a particular rank-reduced transform. Moreover, terms in ${\mathcal T}_p$ are represented as a combination of three operations ${\mathcal F}_k$, ${\mathcal Q}_k$ and ${\boldsymbol{\phi}}_k$ with $k=1,...,p$. The prime idea is to determine ${\mathcal F}_k$ separately, for each $k=1,...,p$, from an associated rank-constrained minimization problem similar to that used in the Karhunen--Lo\`{e}ve transform. The operations ${\mathcal Q}_k$ and ${\boldsymbol{\phi}}_k$ are auxiliary for finding ${\mathcal F}_k$. The contribution of each term in ${\mathcal T}_p$ improves the entire transform performance. A corresponding unconstrained nonlinear optimal transform is also considered. Such a transform is important in its own right because it is treated as an optimal filter without signal compression. A rigorous analysis of errors associated with the proposed transforms is given., Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m by n matrix A by a matrix of rank k at most., Comment: 4 pages