signal processing teams

Abstract: Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\rightarrow \infty, 2n/d^2 \rightarrow α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \in \{-1,+1\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for an explicit expression of the standard deviation $σ(α)<1$ for all $0<α<α_* \simeq 5.74$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025) in this interval.

Abstract: Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator \(M=JJ^\ast\). Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator \(B\), and integrating over local fluctuations yields $$ Φ_{\mathrm{fluc}}(M;B)=\frac{σ_ξ^2}{2}\log\det(M^{-1}+B)+\mathrm{const}. $$ At fixed spectrum, this term is rotationally stationary when \([M,B]=0\), is minimized by pairing large eigenvalues of \(M\) with small eigenvalues of \(B\), and generates a local restoring contribution against rotational mismatch. For ReLU-type function spaces under mild stable statistical conditions, \(B=σ_ξ^2L^\ast\mathcal K L\), where \(L\) measures coarse-grained second-order structure. Thus the low-\(B\) sector corresponds, up to bounded anisotropy of \(\mathcal K\), to low structural curvature, implying a preference for faster relaxation along smooth, data-adaptive directions. These results identify function space as a natural macroscopic level for studying stable collective organization in learning.

Abstract: This work addresses a hardware constraint in reservoir computing: the limited size of the readout layer imposed by systems with a physical readout. We investigate a strategy to accommodate this constraint based on random projection, which compresses high-dimensional reservoir states into a lower-dimensional subspace while preserving key properties of the source space and information- processing capabilities. To evaluate this approach, we compare a small, standalone time delay reservoir against a larger configuration whose output is projected down to match the same restricted readout dimension. Using task-independent metrics, we demonstrate that the distribution of information-processing capacities may differ between the two configurations, even at identical readout sizes. Furthermore, we perform a comprehensive hyperparameter scan to assess how both systems behave under varying physical regimes. Finally, we benchmark this approach on the standard NARMA10 task, showing that the random projection framework can yield superior performance compared to a standalone constrained reservoir, within specific compression range. These results provide a scalable pathway to bypass physical readout bottlenecks in hardware-based reservoir computing.