Norm Distortion Research Articles

Robustness properties of dimensionality reduction with Gaussian random matrices

In this paper, motivated by the results in compressive phase retrieval, we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal restricted isometry property (RIP) constants, which plays a central role in the study of phaseless compressed sensing. As a byproduct of our results, we also establish the robustness property of Gaussian random finite frames under erasure.

Homomorphisms with small bound between Fourier algebras

Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism $T$ between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm $\| T \|_{cb} < \sqrt {3/2}$ (resp. $ \| T \|_{cb} < \sqrt {5}/2$). We show similar results involving the norm distortion $\| T \| \| T ^{-1} \|$ with universal but non-explicit bound. Our results subsume Walter's well-known structural theorems and also Lau's theorem on second conjugate of Fourier algebras.

The High Resolution Vector Quantization Problem with Orlicz Norm Distortion

We derive a high-resolution formula for the quantization problem under Orlicz norm distortion. In this setting, the optimal point density solves a variational problem which comprises a function g:ℝ+→[0,∞) characterizing the quantization complexity of the underlying Orlicz space. Moreover, asymptotically optimal codebooks induce a tight sequence of empirical measures. The set of possible accumulation points is characterized, and in most cases it consists of a single element. In that case, we find convergence as in the classical setting.

The Coding Complexity of Lévy Processes

We introduce a new coding scheme for general real-valued Levy processes and control its performance with respect to L p [0,1]-norm distortion under different complexity constraints. We also establish lower bounds that prove the optimality of our coding scheme in many cases.

The coding complexity of diffusion processes under supremum norm distortion

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under supremum norm distortion. Tight asymptotic formulas are found under mild regularity assumptions. The main technical tool is a decoupling method which allows us to relate the complexity of the diffusion process to that of the Wiener process. The technique is also applicable when considering the L p [ 0 , 1 ] -norm distortion.

The coding complexity of diffusion processes under [formula omitted]-norm distortion

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under L p [ 0 , 1 ] -norm distortion. We find explicit high resolution formulas in terms of the average diffusion coefficient seen by the process. The proof is based on a decoupling method introduced in a former article by the author. Given that link it remains to analyze the coding problem for a concatenation of Wiener processes and to solve the corresponding rate allocation problem.

Random ε-nets and embeddings in lN∞

We show that, given an n-dimensional normed space X, a sequence of N = (8/e) 2n independent random vectors (Xi) N=1, uniformly distributed in the unit ball of X � , with high probability forms an e-net for this unit ball. Thus the random linear map Γ : R n → R N defined by Γ x = (h x, Xii ) N=1 embeds X in l N with at most 1 + e norm distortion. In the case X = l n we obtain a random 1 + e-embedding into l N with asymptotically best possible relation between N, n, and e. 1. Introduction. Let X = (R n , kk ) be an arbitrary n-dimensional normed space with unit ball K. It is well known that, for any 0 < e < 1, X can be 1 + e-embedded into l N, for some N = N(e, n), depending on e and n, but independent of X. In this note we investigate 1+e-isomorphic embed- dings which are random with respect to some natural measure, depending on K. We first show that for N = (8/e) 2n , a sequence of N independent ran- dom vectors (Xi) N=1 , uniformly distributed in the unit ball K 0 of the dual space X ∗ , forms an e-net for K 0 with high probability. Thus, with high prob- ability, the random linear map Γ : R n → R N defined by Γ x = (h x, Xii ) N=1 embeds X in l N with at most 1 + e norm distortion. The important case is X = l n . In this case it is more natural to consider random vectors Xi uniformly distributed on the sphere S n−1 . Such vectors also form an e-net on the sphere, hence they determine a random 1 + e- embedding Γ of l n into l N. We also show that p n/N Γ is a 1+ e-isometry from l n into l N , with high probability. The case X = l n is connected with Dvoretzky's theorem ((D)). Milman found a new proof ((M)), using the Levy isoperimetric inequality on the

A Partial Norm Based Early Rejection Algorithm for Fast Motion Estimation

Recently, many algorithms have been proposed for fast full search motion estimation. Among them, successive elimination algorithm (SEA) and its modified algorithms significantly speed up the performance of the full search algorithm. By introducing the inequality equation between the norm and the mean absolute difference (MAD) of two matching blocks, the SEA can successively eliminate invalid candidate blocks without any loss in estimation accuracy. In this paper, we propose a partial norm based early rejection algorithm (PNERA) for fast block motion estimation. The proposed algorithm employs the sum of partial norms from several subblocks of the block. Applying the sum of partial norms to the inequality equation, we can significantly reduce the computational complexity of the full search algorithm. In an attempt to reduce the computational load further, the modified algorithms using partial norm distortion elimination (PNDE) and subsampling methods are also proposed. Experimental results show that the proposed algorithm is about 4 to 9 times faster than the original exhaustive full search, and is about 3 to 4 times faster than the SEA.

Recursive coding of spectrum parameters

A theoretical analysis of recursive speech spectrum coding, where predictive and finite state schemes are special cases, is presented. We evaluate the spectral distortion (SD) theoretically and design coders that minimize the SD. The analysis rests on three cornerstones: high-rate theory, PDF modeling, and an approximation of SD. A derivation of the mean L/sub 2/-norm distortion of a recursive quantizer operating at high rate is provided. Also, the distortion distribution is supplied. The evaluation of the distortion expressions requires a model of the joint PDF of two consecutive spectrum vectors. The LPC spectrum source considered here has outcomes in a bounded region, and this is taken into account in the choice of model and modeling algorithm. It is further shown how to approximate the SD with an L/sub 2/-norm measure. Combining the results, we show theoretically that 16 bits are needed to achieve an average SD of 1 dB when quantizing ten-dimensional (10-D) spectrum vectors using a first-order recursive scheme. A gain of six bits per frame is noted compared to memoryless quantization. These results rely on high-rate assumptions which are validated in experiments. There, actual high-rate optimal coders are designed and evaluated.