The Adversarial Robustness of Sampling

Random sampling is a fundamental primitive in modern algorithms, statistics, and machine learning, used as a generic method to obtain a small yet "representative" subset of the data. In this work, we investigate the robustness of sampling against adaptive adversarial attacks in a streaming setting: An adversary sends a stream of elements from a universe $U$ to a sampling algorithm (e.g., Bernoulli sampling or reservoir sampling), with the goal of making the sample "very unrepresentative" of the underlying data stream. The adversary is fully adaptive in the sense that it knows the exact content of the sample at any given point along the stream, and can choose which element to send next accordingly, in an online manner. Well-known results in the static setting indicate that if the full stream is chosen in advance (non-adaptively), then a random sample of size $\Omega(d / \varepsilon^2)$ is an $\varepsilon$-approximation of the full data with good probability, where $d$ is the VC-dimension of the underlying set system $(U,R)$. Does this sample size suffice for robustness against an adaptive adversary? The simplistic answer is \emph{negative}: We demonstrate a set system where a constant sample size (corresponding to VC-dimension $1$) suffices in the static setting, yet an adaptive adversary can make the sample very unrepresentative, as long as the sample size is (strongly) sublinear in the stream length, using a simple and easy-to-implement attack. However, this attack is "theoretical only", requiring the set system size to (essentially) be exponential in the stream length. This is not a coincidence: We show that to make Bernoulli or reservoir sampling robust against adaptive adversaries, the modification required is solely to replace the VC-dimension term $d$ in the sample size with the cardinality term $\log |R|$. This nearly matches the bound imposed by the attack

Hierarchical Interpretation of Fractal Image Coding and Its Applications

The basics of a block oriented fractal image coder are reviewed. The output of the coder is an IFS (Iterated Function System), which approximates the image as a fixed point of a contractive transformation. A new hierarchical interpretation of the IFS code, which relates the different scales of the fixed point, is introduced. We prove the existence of a unique function of a continuous variable that is associated with the IFS code. It is further shown that the different scales of the IFS fixed point are directly computable from this so called IFS embedded function. The computation of the IFS-code, depends on the sampling method, an issue that is also discussed. A matrix representation of the IFS code is described and related to the fractal dimension of the IFS embedded function. An application to a new super-resolution method, using an IFS-code, is demonstrated, and its characteristics are analyzed. Another application of the hierarchical representation to fast decoding is also presented..

A Multi-Resolution Framework for Fractal Image Representation and its applications

The starting point of this paper is the basic fractal coder suggested by Jacquin. The coder finds and encodes the parameters of a partitioned iterated function system (PIFS), which approximates the signal as a fixed-point of a contractive transformation. The work presented here can be divided into two parts. The first part begins with a presentation of the hierarchical structure of the PIFS code. This structure relates the code and its fixed-point in different resolutions. It is shown that there exists a function of a continuous variable which is directly related to the PIFS. It is shown that by properly manipulating this function, called the PIFS embedded-function, one can compute the fixed-points related to the code in any desired resolution. We end the first part with a brief description of several applications, such as a fast non-iterative decoder, a method for fractal interpolation of the signal via its PIFS code, and an improved collage-bound. This research was supported by the ..