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Abstract. We consider the problem of reconstructing an N -dimensional continuous vector x from P constraints which are generated by its linear transformation under the assumption that the number of non-zero elements of x is typically limited to ρN (0 ≤ ρ ≤ 1). Problems of this type can be solved by minimizing a cost function with respect to the L p -norm ||x|| p = lim ǫ→+0 N i=1 |x i | p+ǫ , subject to the constraints under an appropriate condition. For several p, we assess a typical case limit α c (ρ), which represents a critical relation between α = P/N and ρ for successfully reconstructing the original vector by minimization for typical situations in the limit N, P → ∞ with keeping α finite, utilizing the replica method. For p = 1, α c (ρ) is considerably smaller than its worst case counterpart, which has been rigorously derived by existing literature of information theory.

Abstract-Channel polarization is a method of constructing capacity achieving codes for symmetric binary-input discrete memoryless channels (B-DMCs) [1]. In the original paper, the construction complexity is exponential in the blocklength. In this paper, a new construction method for arbitrary symmetric binary memoryless channel (B-MC) with linear complexity in the blocklength is proposed. Furthermore, new upper bound and lower bound of the block error probability of polar codes are derived for the BEC and arbitrary symmetric B-MC, respectively.

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I present a mean-field theory for Boltzmann machine learning, derived by employing Thouless-Anderson-Palmer free energy formalism to a full extent. Using the Plefka expansion an extended theory that takes higher-order correction to mean-field free energy formalism into consideration is presented, from which the mean-field approximation of general orders, along with the linear response correction, are derived by truncating the Plefka expansion up to desired orders. A theoretical foundation for an effective trick of using ''diagonal weights,'' introduced by Kappen and Rodríguez, is also given. Because of the finite system size and a lack of scaling assumptions on interaction coefficients, the truncated free energy formalism cannot provide an exact description in the case of Boltzmann machines. Accuracies of mean-field approximations of several orders are compared by computer simulations.