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Differential privacy (DP) is a leading framework for privacy protection, yet privacy mechanisms tailored to functional data analysis (FDA) remain relatively underexplored. For functional mean estimation, we investigate a fundamental question regarding the ordering of privatization and estimation by comparing two canonical pipelines: Fit-Then-Protect (FTP), which smooths trajectories before privatization, and Protect-Then-Fit (PTF), which privatizes observations at discrete time points before smoothing. When data are observed without error, we prove that FTP achieves strictly smaller mean squared error than PTF asymptotically (as the sample size increases) under the same $μ$-Gaussian differential privacy ($μ$-GDP) budget in both finite-dimensional Hilbert spaces and infinite-dimensional reproducing kernel Hilbert spaces (RKHS). Simulation studies support our theory and explore settings with observation error and fitting constraints, demonstrating that FTP outperforms PTF under small observation noise, whereas PTF can become preferable in high-noise settings. In addition to this comparison, we establish a novel GDP guarantee for Gaussian perturbations taking values in a Banach space, extending existing results beyond the RKHS setting.

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as budgeted heteroskedastic multi-judge estimation under independent, unbiased score observations. Given $K$ prompt-response pairs and $J$ judges with known costs but unknown query-judge variances, our goal is to estimate the bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its leading error bound. Since this allocation depends on the unknown variances, we then address the unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and uses optimistic variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a local minimax lower bound matching the leading allocation-dependent rate, establishing asymptotic instance-optimality up to logarithmic factors. An Assouad-type construction based on local, variance-preserving perturbations yields this bound both in expectation and with constant probability. A complementary Fano-type change-of-measure argument captures the logarithmic dependence on the confidence level with a different (suboptimal) allocation geometry. Finally, we numerically validate our approach on synthetic data and resampled HelpSteer2 evaluations, demonstrating improvements over naïve uniform allocation.

We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik-Ben Arous-Peche (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions.

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliably when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of setting.

Conformal prediction gives finite-sample coverage when calibration and test points are exchangeable, but exchangeability can fail when deployment changes the data distribution. We study conformal prediction in performative settings, where the deployed predictor affects the data distribution, and seek prediction sets valid at performative equilibrium: coverage must hold under $D(f^\star)$ for a performatively stable predictor $f^\star$. Under total-variation $L$-sensitivity of the distribution map, we prove a phase transition at $L=1$. If $L \ge 1$, no finite-sample procedure can uniformly guarantee both equilibrium-valid coverage and nontrivial prediction sets. If $L<1$, the minimax excess miscoverage over informative procedures is $\widetildeΩ(1/((1-L)\sqrt{n}))$, implying a sample-complexity inflation of order $1/(1-L)^2$ for fixed excess miscoverage. Our main technical tool is a transfer theorem: any procedure returning a predictor within $\varepsilon$ of equilibrium yields equilibrium-valid coverage up to an additive $L\varepsilon$ term after split-conformal calibration. We instantiate the theorem with two retraining algorithms: a model-free method for discretizable classes and a parametric tail-averaged method that matches the $n$- and $L$-dependence of this rate up to logarithmic factors. Experiments, including semi-synthetic credit scoring, demonstrate the predicted threshold behavior and scaling laws.