This paper is about private data analysis, in which a trusted curator holding a confidential database responds to real vector-valued queries. A common approach to ensuring privacy for the database elements is to add appropriately generated random noise to the answers, releasing only these noisy responses. A line of study initiated in [7] examines the amount of distortion needed to prevent priva...

1 Background The theory of holomorphic motions, introduced by Mané-Sad-Sullivan [1] and advanced by Slodkowski [2], has had a significant impact on the theory of quasiconformal mappings. A reasonably thorough account of this is given in our book [3]. In [4, 5] we established some classical distortion theorems for quasiconformal mappings and used the theory to develop connections between Schottk...

We derive a simple general parametric representation of the rate–distortion function of a memoryless source, where both the rate and the distortion are given by integrals whose integrands include the minimum mean square error (MMSE) of the distortion ∆ = d(X, Y ) based on the source symbol X , with respect to a certain joint distribution of these two random variables. At first glance, these rel...

The Shannon lower bound is one of the few lower bounds on the rate-distortion function that holds for a large class of sources. In this paper, it is demonstrated that its gap to the rate-distortion function vanishes as the allowed distortion tends to zero for all sources that have a finite differential entropy and whose integer parts have a finite entropy. Conversely, it is demonstrated that if...

We consider the separate source coding problem of correlated Gaussian observations . We consider the case that satisfy where are correlated Gaussian random variables and are independent additive Gaussian noises also independent of . On this coding system the determination problem of the rate distortion region remains open. In our previous work, we derived explicit outer and inner bounds of the ...

This paper considers the quantization problem on the Grassmann manifold with dimension n and p. The unique contribution is the derivation of a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results are only valid for either p = ...

This paper presents an overview of existing bounds on l1-embeddings of planar metrics. A new family of graphs containing the K2,3 minor is introduced. Computational results on this family of graphs establish a new lower bound on the constant in the following conjecture: There exists an absolute constant Cmax > 0 such that every finite planar metric embeds into the l1 metric with distortion < Cm...

In this paper, we present new bounds for the rate loss of multiresolution source codes (MRSCs). Considering an -resolution code, the rate loss at the th resolution with distortion is defined as = ( ), where is the rate achievable by the MRSC at stage . This rate loss describes the performance degradation of the MRSC compared to the best single-resolution code with the same distortion. For two-r...

Rate-distortion theory is applied to the problem of joint compression and classification. A Lagrangian distortion measure is used to consider both the squared Euclidean error in reconstructing the original data as well as the classification performance. The bound is calculated based on an alternating-minimization procedure, representing an extension of the Blahut-Arimoto algorithm. As an exampl...