Shapley value is a widely used tool in explainable artificial intelligence (XAI), as it provides a principled way to attribute contributions of input features to model outputs. However, estimation of Shapley value requires capturing conditional dependencies among all feature combinations, which poses significant challenges in complex data environments. In this article, EmSHAP (Energy-based model for Shapley value estimation), an accurate Shapley value estimation method, is proposed to estimate the expectation of Shapley contribution function under the arbitrary subset of features given the rest. By utilizing the ability of energy-based model (EBM) to model complex distributions, EmSHAP provides an effective solution for estimating the required conditional probabilities. To further improve estimation accuracy, a GRU (Gated Recurrent Unit)-coupled partition function estimation method is introduced. The GRU network captures long-term dependencies with a lightweight parameterization and maps input features into a latent space to mitigate the influence of feature ordering. Additionally, a dynamic masking mechanism is incorporated to further enhance the robustness and accuracy by progressively increasing the masking rate. Theoretical analysis on the error bound as well as application to four case studies verified the higher accuracy and better scalability of EmSHAP in contrast to competitive methods.

ReLU is a widely used activation function in deep neural networks. This paper explores the stability properties of the ReLU map. For any weight matrixA∈ Rm×nand bias vectorb∈ Rmat a given layer, we define the condition number κA,bas κA,b=UA,b/LA,b, whereUA,bandLA,bare the upper and lower Lipschitz constants, respectively. We first demonstrate that for any given A and b, the condition number satisfies κA,b≥ √ 2. Moreover, when the weights of the network at a given layer are initialized as random i.i.d. Gaussian variables and the bias term is set to zero, the condition number asymptotically approaches this lower bound. Our findings offer valuable insights into the characteristics of randomly initialized neural networks, contributing to a better understanding of their initial behavior and potential performance.

In this paper, we consider the sparse phase retrieval problem, that is, recovering an unknown$s$-sparse signal$\boldsymbol {x}\in \mathbb {R}^{n}$from intensity-only measurements. Specifically, we focus on the problem of recovering$\boldsymbol {x}$from the observations that are cyclically convoluted with a kernel$\boldsymbol {a}\in \mathbb {R}^{n}$, denoted as$\boldsymbol {y}=|R_{I}(\boldsymbol {a}\circledast \boldsymbol {x})|^{2}$. Here,$R_{I}: \mathbb {R}^{n}\rightarrow \mathbb {R}^{|I|}$represents a random subsampled operator that is restricted to the index set$I$. This model is motivated by real-world applications in optics and communications. We provide that if$\boldsymbol {a}$is a random Gaussian vector and the number of subsampled measurements is on the order of$s\text {polylog}(n)$, one can recover$\boldsymbol {x}$up to a global phase provided that the initialization estimator is around$\boldsymbol {x}$. It is the first theoretical result that discusses the behavior of sparse convolutional phase retrieval under physically realistic measurements, as opposed to independent Gaussian measurements.

This paper introduces the restricted eigenvalue condition adapted to frame D (D-RE), which is a natural extension to the standard restricted eigenvalue condition. The D-RE condition is a relaxation of the D†-RIP, where D†= (DD*)-1D is the canonical dual frame of D. We establish the D-RE condition for several classes of correlated measurement matrices, when the covariance matrix of row measurements satisfies the D-RE condition. Furthermore, by the D-RE condition, we get the error bounds in the analysis LASSO (ALASSO) and the analysis Dantzig Selector (ADS) under a sparsity scenario. In order to recover non-sparse signals, we consider the robust 12 D-nullspace property of correlated Gaussian matrices. Similarly, we get the error estimations in the ALASSO and the ADS in non-sparse case. The approximation equivalence between the ALASSO and the ADS is also established by calculating prediction loss difference.