Motivated by the geometric advantages of quaternions in representing rotations and postures, we propose a quaternion-valued supervised learning Hopfield-structured neural network (QSHNN) with a fully connected structure inspired by the classic Hopfield neural network (HNN). Starting from a continuous-time dynamical model of HNNs, we extend the formulation to the quaternionic domain and establish the existence and uniqueness of fixed points with asymptotic stability. For the learning rules, we introduce a periodic projection strategy that modifies standard gradient descent by periodically projecting each 4 4block of the weight matrix onto the closest quaternionic structure in the least-squares sense. This approach preserves both convergence and quaternionic consistency throughout training. Benefiting from this rigorous mathematical foundation, the experimental model implementation achieves high accuracy, fast convergence, and strong reliability across randomly generated target sets. Moreover, the evolution trajectories of the QSHNN exhibit well-bounded curvature, i.e., sufficient smoothness, which is crucial for applications such as control systems or path planning modules in robotic arms, where joint postures are parameterized by quaternion neurons. Beyond these application scenarios, the proposed model offers a practical implementation framework and a general mathematical methodology for designing neural networks under hypercomplex or non-commutative algebraic structures.

Denoiser models have become powerful tools for inverse problems, enabling the use of pretrained networks to approximate the score of a smoothed prior distribution. These models are often used in heuristic iterative schemes aimed at solving Maximum a Posteriori (MAP) optimisation problems, where the proximal operator of the negative log-prior plays a central role. In practice, this operator is intractable, and practitioners plug in a pretrained denoiser as a surrogate--despite the lack of general theoretical justification for this substitution. In this work, we show that a simple algorithm, closely related to several used in practice, provably converges to the proximal operator under a log-concavity assumption on the prior p. We show that this algorithm can be interpreted as a gradient descent on smoothed proximal objectives. Our analysis thus provides a theoretical foundation for a class of empirically successful but previously heuristic methods.

A panel satisfies descriptive representation when its composition reflects the population. We examine the role of descriptive representation in collective decision making through an optimization lens, asking whether representative panels make decisions that maximize social welfare for the underlying population. Our main results suggest that, in general, representation with respect to intersections of two or more features guarantees higher social welfare than that achieved by the status quo of proportionally representing individual features. Moreover, an analysis of real data suggests that representation with respect to pairs of features is feasible in practice. These results have significant implications for the design of citizens' assemblies, which are gaining prominence in AI governance.

Generative modeling of non-negative, discrete data, such as symbolic music, remains challenging due to two persistent limitations in existing methods. First, most approaches rely on modeling continuous embeddings, which are not wellsuited for inherently discrete data distributions.

The influence function (IF) of a statistical functional is the Riesz representer of its derivative, also known as its first variation and Fisher-Rao gradient. It is a key object for numerical optimization over probability measures, semiparametric efficiency theory, standard constructions of efficient estimators, and an arsenal of inference methods for these estimators. Yet, deriving the IF analytically is often an obstruction for practitioners. To automate this task, we develop a novel spectral representation of the IF that lends itself to a low-rank functional estimator in a reproducing kernel Hilbert space (rkHs). Our estimator (i) does not require analytic derivations by the user, (ii) relies on kernel Principal Component Analysis and numerical pathwise derivatives along these components. We present the derivation of the representation and prove consistency of the low-rank rkHs estimator.

We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on n data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order log(n)neurons suffices for global convergence with high probability. Our analysis uses a Polyak-Łojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of 1n. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders 1n and 1 n, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order 1log(n).

We propose and analyze a conservative drifting method for one-step generative modeling. The method replaces the original displacement-based drifting velocity by a kernel density estimator (KDE)-gradient velocity, namely the difference of the kernel-smoothed data score and the kernel-smoothed model score. This velocity is a gradient field, addressing the non-conservatism issue identified for general displacement-based drifting fields. We prove continuous-time finite-particle convergence bounds for the conservative method on $\R^d$: a joint-entropy identity yields bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction is a reciprocal-KDE self-interaction term, and we give deterministic and high-probability local-occupancy conditions under which this term is controlled. We keep the quadrature constants explicit and track their possible bandwidth dependence: the root residual-velocity rate $N^{-1/(d+4)}$ holds under an additional $h$-uniform quadrature regularity condition, while a more general growth condition yields the optimized root rate $N^{-(2-β)/(2(d+4-β))}$, where $0\le β<2$. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in Deng et al., 2026 (arxiv:2602.04770). For this method, a sharp companion kernel decomposes the velocity into a positive scalar preconditioning of a sharp-score mismatch plus a Laplace scale-mismatch residual, producing an analogous finite-particle rate with an unavoidable residual term. Finally, we explain how the continuous-time residual-velocity bounds translate into one-step generation guarantees through the explicit drift size $η$.

We develop a gradient flow on the space of probability measures defined on matrix-valued parameters induced by regularized Muon, an analytically smoothed version of the idealized Muon optimizer. The key observation is that the regularized orthogonalization map is the gradient of a smooth Fenchel-dual smoothing of the nuclear norm. This identifies the (regularized) Muon update as a mirror/prox step in the update variable, with momentum acting as the dual coordinate. We use this structure to lift Muon from a single matrix parameter to finite-particle probability objectives of the form $J(ρ)=R\left(\int F d ρ\right)$, a setting motivated by mean-field descriptions of neural-network training, and derive the inertial continuous-time limit. Using this structure, we derive the finite-particle continuous-time limit under the inertial scaling of step size and momentum, and then pass to a phase-space mean-field equation over probability laws on parameter-momentum pairs. The resulting flow can be shown to be a damped Hamiltonian probability dynamics whose kinetic energy is induced by the regularized Muon mirror potential. We prove an exact Hamiltonian dissipation identity, showing that the Hamiltonian energy decreases monotonically. While the target objective itself need not be monotone along the inertial Muon dynamics, under additional gradient-dominance, bounded-momentum, and curvature/alignment assumptions, we obtain continuous and discrete-time exponential convergence rates for the objective gap. We also study the well-posedness of the mean-field limit equation and establish propagation of chaos guarantees for the interacting particle system. Finally, we extend the formulation to Hilbert-valued feature maps on product matrix spaces, yielding a blockwise Muon probability flow applicable to smooth transformer mixture-of-experts models.

Physics-Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high-dimensional space, enabling the use of Koopman-based analysis and showing that high-rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.

We study the large-depth limit of transformers trained with AdamW, by modelling the hidden-state dynamics as an interacting particle system (IPS) coupled through the attention mechanism. Under appropriate scaling of the attention heads, we prove that the joint dynamics of the hidden states and backpropagated variables converge in $L^2$, uniformly over the initial condition, to the solution of a forward--backward system of ODEs at rate $\mathcal O(L^{-1}+L^{-1/3}H^{-1/2})$. Here, $L$ and $H$ denote the depth and number of heads of the transformer, respectively. The limiting system of ODEs can be identified with a McKean--Vlasov ODE (MVODE) when the attention heads do not incorporate causal masking. By using the flow maps associated with this MVODE and applying concentration of measure techniques, we obtain bounds on the difference between the discrete and continuous models that are uniform over compact sets of initial conditions. As this is achieved without resorting to a covering argument, the constants in our bounds are independent of the number of tokens. Furthermore, under a suitable adaptation to AdamW, the bounds become independent of the token embedding dimension.