Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a small number of noisy linear measurements is an important problem in compressed sensing. In this paper, the high-dimensional setting is considered. It is shown that if the measurement rate and per-sample signal-to-noise ratio (SNR) are finite constants independent of the length of the vector, then the optimal sparsity pattern estimate will have a constant fraction of errors. Lower bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector. The tightness of the bounds in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing achievable bounds. Near optimality is shown for a wide variety of practically motivated signal models

The pattern of recovery

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A study of pattern recovery in recurrent correlation associative memories

In this paper, we analyze the recurrent correlation associative memory (RCAM) model of Chiueh and Goodman. This is an associative memory in which stored binary memory patterns are recalled via an iterative update rule. The update of the individual pattern-bits is controlled by an excitation function, which takes as its arguement the inner product between the stored memory patterns and the input patterns. Our contribution is to analyze the dynamics of pattern recall when the input patterns are corrupted by noise of a relatively unrestricted class. We make three contributions. First, we show how to identify the excitation function which maximizes the separation (the Fisher discriminant) between the uncorrupted realization of the noisy input pattern and the remaining patterns residing in the memory. Moreover, we show that the excitation function which gives maximum separation is exponential when the input bit-errors follow a binomial distribution. Our second contribution is to develop an expression for the expectation value of bit-error probability on the input pattern after one iteration. We show how to identify the excitation function which minimizes the bit-error probability. However, there is no closed-form solution and the excitation function must be recovered numerically. The relationship between the excitation functions which result from the two different approaches is examined for a binomial distribution of bit-errors. The final contribution is to develop a semiempirical approach to the modeling of the dynamics of the RCAM. This provides us with a numerical means of predicting the recall error rate of the memory. It also allows us to develop an expression for the storage capacity for a given recall error rate