The central objective of empirical asset pricing is to identify firm-level signals that explain the cross-section of expected stock returns--whether through exposure to risk factors or persistent mispricing. The dominant paradigm, grounded in the assumption of self-predictability, asserts that a firm's own characteristics forecast its own returns (see, e.g., Cochrane (2011); Harvey et al. (2016)). Complementing this view is a growing literature on cross-predictability--the idea that the characteristics or returns of one asset can help forecast the returns of others (see, e.g., Lo and MacKinlay (1990); Hou (2007); Cohen and Frazzini (2008); Cohen and Lou (2012); Huang et al. (2021, 2022)). A key mechanism underpinning this phenomenon is the presence of lead-lag effects, whereby price movements or information from one firm precede and predict those of related firms. Such effects can stem from staggered information diffusion, peer influence within industries, supply chain linkages, or correlated trading by institutional investors that induces price pressure across related assets. Despite recent methodological advances in modeling cross-stock predictability, several foundational questions remain unresolved. Chief among them is how a mean-variance investor can analytically integrate multiple predictive signals when returns are interconnected across assets. Equally crucial is developing a framework that jointly captures both the relevance of individual signals and the structure of return spillovers--enhancing portfolio performance while preserving interpretability .

BytheMarkovian assumption forlatent state vectors, the Hessian matrix is tri-block diagonal. To facilitate convergence, we initialize the Newton update with a smoothing estimate bylocalGaussian approximation. Theforwardfiltering foradynamic Poisson modelhas been previously described (Eden etal., 2004), and we use anadditional backward pass tosmooth (Rauchetal.,1965). Without constraints, the sampling ofh(j), g(j) and σ2(j) is the same as shown previously. The update of A(j), b(j) and Q(j) is the standard multivariate Bayesian linear regression.

LargeNumberofNodes In this section, we test DAGMA, GOLEM, and NOTEARS for graphs with number of variables d {200,300,500,800,1000}.

Once closed-form expressions for these Jacobians are derived, it remains to substitute those expressions into (16). The following identity (often termed the "vec" rule) will To depict the spatial topographies of the latent components measured on the EEG and fMRI analyses, the "forward-model" [ The results of the comparison are shown in Fig S1, where it is clear that the signal fidelity of the GCs (right panel) significantly exceeds those yielded by PCA (left) and ICA (middle). GCA is only able to recover sources with temporal dependencies (i.e., s Both the single electrodes and Granger components exhibit two pronounced peaks in the spectra: one near 2 Hz ("delta" Fig S3 shows the corresponding result for the left motor imagery condition. EEG motor imagery dataset described in the main text. For each technique, the first 6 components are presented.

Suppose h ( )=e x p ( ) . M -matrix (see [ 50 ]).

By analyzing the loss landscape of a single Transformer layer using Softmax and Gaussian attention kernels, our work provides concrete answers to these questions.

However, there is a noticeable gap in analysis for multiclass classification, with only a handful of results which themselves are restricted to the cross-entropy loss.

The Gromov-Wasserstein (GW) distance is an extension of the optimal transport problem that allows one to match objects between incomparable spaces. At its core, the GW distance is specified as the solution of a non-convex quadratic program and is not known to be tractable to solve. In particular, existing solvers for the GW distance are only able to find locally optimal solutions. In this work, we propose a semi-definite programming (SDP) relaxation of the GW distance. The relaxation can be viewed as the Lagrangian dual of the GW distance augmented with constraints that relate to the linear and quadratic terms of transportation plans. In particular, our relaxation provides a tractable (polynomial-time) algorithm to compute globally optimal transportation plans (in some instances) together with an accompanying proof of global optimality. Our numerical experiments suggest that the proposed relaxation is strong in that it frequently computes the globally optimal solution. Our Python implementation is available at https://github.com/tbng/gwsdp.