The Fisher Information Matrix (FIM) is the canonical local measure of the curvature of a statistical model's log-likelihood surface, and its dominant eigenvalue λmax quantifies the worst-case sensitivity of the model's output distribution to infinitesimal parameter perturbation [1, 2]. The spectral properties of the FIM of neural networks have been studied directly in the random matrix theory literature. Pennington and Worah [4] derive the limiting spectral density of the FIM of a single-hidden-layer network in the high-dimensional asymptotic regime, building on the broader programme of analysing neural network Hessian and kernel spectra via random matrix methods [5, 6], with subsequent work extending these techniques to deeper architectures and non-asymptotic regimes [7, 8]. These results characterize the typical (bulk and edge) spectral behaviour of the FIM for a fixed network and a random or structured input ensemble. This paper studies a complementary question, posed as a perturbation problem rather than an asymptotic-spectrum problem: how does the dominant eigenvalue of a fixed, evaluated empirical FIM change under two specific structured perturbations of the underlying distribution? The first perturbation is a change in the conditioning input away from a reference (in-distribution) ensemble. The second is a structured additive perturbation of the model's own parameters by finite-precision quantization noise -- a perturbation of independent mathematical interest, since it falls outside the i.i.d.-input asymptotic regime treated in the random matrix literature cited above, and instead concerns a fixed network whose parameters, not its input distribution, are perturbed by a noise process with a specific, analytically tractable structure (Definition 4.1). To our knowledge, this parameterperturbation question for the FIM's dominant eigenvalue, under either source of departure, has not been previously formalized.

Recent empirical work shows that semantically equivalent paraphrases can fool financial sentiment classifiers: although a paraphrase remains close to the original under a strong reference embedding, it may shift the target model's representation enough to change the predicted class. Existing robustness theory either assumes a single-model threat model or focuses mainly on empirical attack algorithms. We develop a continuous local model of semantic paraphrase perturbations that captures this two-model structure. We show that the worst-case local displacement of the target representation, subject to a proxy-model budget, is governed by the largest generalised eigenvalue of a matrix pencil $(A,B)$ constructed from the Jacobians of the two embedding maps. The resulting attackability index $λ^*(x)$ is intrinsic to the local paraphrase geometry and the chosen embedders, yields a closed-form prediction-flip condition for affine readouts, and supports conservative population and finite-sample attackability certificates. For uniform control over classes of affine readouts, we derive a distribution-free VC bound for binary attackability indicators and a scale-sensitive margin bound based on an attackability-adjusted margin that subtracts a local geometric penalty from the standard classifier margin. We also connect the continuous theory to discrete paraphrase search, identify an asymmetry between successful and unsuccessful finite searches, and give a covering condition under which the discrete and continuous settings agree. Finally, we propose an empirical verification framework using soft-token relaxations and generated paraphrase sets to assess the local eigenvalue geometry, prediction-flip condition, and finite-search approximation on a deployed financial-text classifier.

The rapid advancement of diffusion models has enabled high-fidelity and semantically rich text-to-image generation; however, ensuring fairness and safety remains an open challenge. Existing methods typically improve fairness and safety at the expense of semantic fidelity and image quality. In this work, we propose RespoDiff, a novel framework for responsible text-to-image generation that incorporates a dual-module transformation on the intermediate bottleneck representations of diffusion models. Our approach introduces two distinct learnable modules: one focused on capturing and enforcing responsible concepts, such as fairness and safety, and the other dedicated to maintaining semantic alignment with neutral prompts. To facilitate the dual learning process, we introduce a novel score-matching objective that enables effective coordination between the modules. Our method outperforms state-of-the-art methods in responsible generation by ensuring semantic alignment while optimizing both objectives without compromising image fidelity. Our approach improves responsible and semantically coherent generation by ~20% across diverse, unseen prompts.

We propose a method for non-parametric conditional distribution estimation based on partitioning covariate-sorted observations into contiguous bins and using the within-bin empirical CDF as the predictive distribution. Bin boundaries are chosen to minimise the total leave-one-out Continuous Ranked Probability Score (LOO-CRPS), which admits a closed-form cost function with $O(n^2 \log n)$ precomputation and $O(n^2)$ storage; the globally optimal $K$-partition is recovered by a dynamic programme in $O(n^2 K)$ time. Minimisation of within-sample LOO-CRPS turns out to be inappropriate for selecting $K$ as it results in in-sample optimism. We instead select $K$ by $K$-fold cross-validation of test CRPS, which yields a U-shaped criterion with a well-defined minimum. Having selected $K^*$ and fitted the full-data partition, we form two complementary predictive objects: the Venn prediction band and a conformal prediction set based on CRPS as the nonconformity score, which carries a finite-sample marginal coverage guarantee at any prescribed level $\varepsilon$. The conformal prediction is transductive and data-efficient, as all observations are used for both partitioning and p-value calculation, with no need to reserve a hold-out set. On real benchmarks against split-conformal competitors (Gaussian split conformal, CQR, CQR-QRF, and conformalized isotonic distributional regression), the method produces substantially narrower prediction intervals while maintaining near-nominal coverage.

Infact, the SDPmaybeinexacteven fork =3 (see Section 8). Suppose program to(2) (3) (4) (5) nonemptyfeasibleset.

It is an open question whether the search and decision versions of promise CSPs are equivalent. Most known algorithms for PCSPs solve only their \emph{decision} variant, and it is unknown whether they can be adapted to solve \emph{search} as well. The main approaches, called BLP, AIP and BLP+AIP, handle a PCSP by finding a solution to a relaxation of some integer program. We prove that rounding those solutions to a proper search certificate can be as hard as any problem in the class TFNP. In other words, these algorithms are ineffective for search. Building on the algebraic approach to PCSPs, we find sufficient conditions that imply ineffectiveness for search. Our tools are tailored to algorithms that are characterized by minions in a suitable way, and can also be used to prove undecidability results for meta-problems. This way, we show that the families of templates solvable via BLP, AIP, and BLP+AIP are undecidable. Using the same techniques we also analyze several algebraic conditions that are known to guarantee the tractability of finite-template CSPs. We prove that several meta-problems related to cyclic polymorphims and WNUs are undecidable for PCSPs. In particular, there is no algorithm deciding whether a finite PCSP template (1) admits cyclic a polymorphism, (2) admits a WNU.

Generative models of 3D cardiovascular anatomy can synthesize informative structures for clinical research and medical device evaluation, but face a trade-off between geometric controllability and realism. We propose CardioComposer: a programmable, inference-time framework for generating multi-class anatomical label maps based on interpretable ellipsoidal primitives. These primitives represent geometric attributes such as the size, shape, and position of discrete substructures. We specifically develop differentiable measurement functions based on voxel-wise geometric moments, enabling loss-based gradient guidance during diffusion model sampling. We demonstrate that these losses can constrain individual geometric attributes in a disentangled manner and provide compositional control over multiple substructures. Finally, we show that our method is compatible with a wide array of anatomical systems containing non-convex substructures, spanning cardiac, vascular, and skeletal organs.