A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise

We define a (symmetric key) encryption of a signal $$\mathbf{s}\in {\mathbb {R^N}}$$s∈RN as a random mapping $$\mathbf{s}\mapsto \mathbf y =(y_1,\ldots ,y_M)^T\in \mathbb {R}^M$$s↦y=(y1,…,yM)T∈RM known both to the sender and a recipient. In general the recipients may have access only to images $$\mathbf{y}$$y corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) $$\mu =M/N\ge 1$$μ=M/N≥1 and the signal strength parameter $$R=\sqrt{\sum _i {s_i^2/N}}$$R=∑isi2/N, we consider the problem of reconstructing $$\mathbf{s}$$s from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap $$p_{\infty }\in [0,1]$$p∞∈[0,1] between the original signal and its recovered image (known as ’estimate’) as $$N\rightarrow \infty $$N→∞, for a given (’bare’) noise-to-signal ratio (NSR) $$\gamma \ge 0$$γ≥0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full $$p_{\infty } (\gamma )$$p∞(γ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with $$p_{\infty }>0$$p∞>0 for any $$\mu >1$$μ>1 and any $$\gamma <\infty $$γ<∞, with $$p_{\infty }\sim \gamma ^{-1/2}$$p∞∼γ-1/2 as $$\gamma \rightarrow \infty $$γ→∞. In contrast, for the case of purely quadratic nonlinearity, for any ERP $$\mu >1$$μ>1 there exists a threshold NSR value $$\gamma _c(\mu )$$γc(μ) such that $$p_{\infty }=0$$p∞=0 for $$\gamma >\gamma _c(\mu )$$γ>γc(μ) making the reconstruction impossible. The behaviour close to the threshold is given by $$p_{\infty }\sim (\gamma _c-\gamma )^{3/4}$$p∞∼(γc-γ)3/4 and is controlled by the replica symmetry breaking mechanism.