Optimal Dimension-Free Sampling for Regularized Classification

We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and sigmoid loss, hinge loss, and ReLU loss, as prominent and popular representative examples. In particular, we prove $k^2/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_2/k$ regularization, and $k/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_1/k$ regularization. For $\|\cdot\|_2^2/k$ regularization, the sampling complexity depends mainly on a bounded derivative property: if $|g'(x)|\leq g(x)$, and $g(0)>0$, and $g$ is monotonic or convex, then it admits linear in $k$ sampling complexity; otherwise the general bound is $k^2/\varepsilon^2$. However, if $g(0)=0$, our results indicate that no dimension-free bounds are possible, and even sublinear bounds are ruled out. All upper bounds are complemented by matching lower bounds up to polylogarithmic terms. Moreover, our work relies conceptually and algorithmically on simple uniform or (squared) norm sampling and hereby improves over recent cubic $k^3/\varepsilon^2$ sensitivity sampling bounds of (Alishahi and Phillips, ICML'24). This is achieved by refined arguments involving higher moment bounds and empirical process analyses to avoid overcounting that appears in the de-facto standard VC-dimension and sensitivity framework.

Optimizing Computational-Statistical Runtime for Wasserstein Distance Estimation

Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size $n$ from two underlying random samples. Unfortunately, even in lower dimensional Euclidean space problems $\left( d \in \{2,3\} \right)$, algorithms for Wasserstein distance computation with approximate or exact precision guarantees scale poorly in the runtime as a function of $n$ and the desired precision. In response, we consider the computational-statistical runtime, where the goal is to estimate from samples the Wasserstein distance between potentially smooth measures up to $ε$-additive error in expectation with respect to the sampling; we allow $O(1)$ computational cost for collecting a sample. Towards this, we develop a Sample-Sketch-Solve paradigm where we introduce a regular cartesian grid sketch of the samples. We show that (especially under $α$-Hölder smooth distributions) this can compress the data without increasing asymptotic error, and also regularizes the structure which enables faster exact algorithms. Ultimately, we approximate $W_2^2(P,Q)$ within $ε$ error in $ε^{-\max(2,\frac{d+1+o(1)}{1+α})}$ time for $0 < α< 1$ Hölder smooth distributions $P,Q$ on $(0,1)^{d}$; an optimal $Θ(ε^{-2})$ for $α> 1/2$ when $d=2$ and nearly optimal as $α\to 1$ when $d = 3$.

Convolutional Maximum Mean Discrepancy for Inference in Noisy Data

Modern data analyses frequently encounter settings where samples of variables are contaminated by measurement error. Ignoring measurement noise can substantially degrade statistical inference, while existing correction techniques are often computationally costly and inefficient. Recent advances in kernel methods, particularly those based on Maximum Mean Discrepancy (MMD), have enabled flexible, distribution-free inference, yet typically assume precise data and overlook contamination by measurement error. In this work, we introduce a novel framework for inference with samples corrupted by potentially heteroscedastic noise from a known distribution. Central to our approach is the convolutional MMD (convMMD), which compares distributions after noise convolution and retains metric validity under standard kernel conditions. We establish finite-sample deviation bounds that are unaffected by measurement error and prove an equivalence between testing under noise and kernel smoothing. Leveraging these insights, we introduce a convMMD-based estimator for inference with noisy, heteroscedastic observations. We establish its consistency and asymptotic normality, and provide an efficient implementation using stochastic gradient descent. We demonstrate the practical effectiveness of our approach through simulations and applications in astronomy and social sciences.

Hardness of High-Dimensional Linear Classification

We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only $O(n^d)$ and respectively $\tilde O(1/\varepsilon^d)$ upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and $k$-Sum problems. Our reductions yield matching lower bounds of $\tildeΩ(n^d)$ and respectively $\tildeΩ(1/\varepsilon^d)$ based on Affine Degeneracy testing, and $\tildeΩ(n^{d/2})$ and respectively $\tildeΩ(1/\varepsilon^{d/2})$ conditioned on $k$-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.

Curveball Steering: The Right Direction To Steer Isn't Always Linear

Activation steering is a widely used approach for controlling large language model (LLM) behavior by intervening on internal representations. Existing methods largely rely on the Linear Representation Hypothesis, assuming behavioral attributes can be manipulated using global linear directions. In practice, however, such linear interventions often behave inconsistently. We question this assumption by analyzing the intrinsic geometry of LLM activation spaces. Measuring geometric distortion via the ratio of geodesic to Euclidean distances, we observe substantial and concept-dependent distortions, indicating that activation spaces are not well-approximated by a globally linear geometry. Motivated by this, we propose "Curveball steering", a nonlinear steering method based on polynomial kernel PCA that performs interventions in a feature space, better respecting the learned activation geometry. Curveball steering consistently outperforms linear PCA-based steering, particularly in regimes exhibiting strong geometric distortion, suggesting that geometry-aware, nonlinear steering provides a principled alternative to global, linear interventions.

Understanding and Mitigating Dataset Corruption in LLM Steering

Contrastive steering has been shown as a simple and effective method to adjust the generative behavior of LLMs at inference time. It uses examples of prompt responses with and without a trait to identify a direction in an intermediate activation layer, and then shifts activations in this 1-dimensional subspace. However, despite its growing use in AI safety applications, the robustness of contrastive steering to noisy or adversarial data corruption is poorly understood. We initiate a study of the robustness of this process with respect to corruption of the dataset of examples used to train the steering direction. Our first observation is that contrastive steering is quite robust to a moderate amount of corruption, but unwanted side effects can be clearly and maliciously manifested when a non-trivial fraction of the training data is altered. Second, we analyze the geometry of various types of corruption, and identify some safeguards. Notably, a key step in learning the steering direction involves high-dimensional mean computation, and we show that replacing this step with a recently developed robust mean estimator often mitigates most of the unwanted effects of malicious corruption.

Efficient and Stable Multi-Dimensional Kolmogorov-Smirnov Distance

We revisit extending the Kolmogorov-Smirnov distance between probability distributions to the multidimensional setting and make new arguments about the proper way to approach this generalization. Our proposed formulation maximizes the difference over orthogonal dominating rectangular ranges (d-sided rectangles in R^d), and is an integral probability metric. We also prove that the distance between a distribution and a sample from the distribution converges to 0 as the sample size grows, and bound this rate. Moreover, we show that one can, up to this same approximation error, compute the distance efficiently in 4 or fewer dimensions; specifically the runtime is near-linear in the size of the sample needed for that error. With this, we derive a delta-precision two-sample hypothesis test using this distance. Finally, we show these metric and approximation properties do not hold for other popular variants.

No Dimensional Sampling Coresets for Classification

We refine and generalize what is known about coresets for classification problems via the sensitivity sampling framework. Such coresets seek the smallest possible subsets of input data, so one can optimize a loss function on the coreset and ensure approximation guarantees with respect to the original data. Our analysis provides the first no dimensional coresets, so the size does not depend on the dimension. Moreover, our results are general, apply for distributional input and can use iid samples, so provide sample complexity bounds, and work for a variety of loss functions. A key tool we develop is a Radamacher complexity version of the main sensitivity sampling approach, which can be of independent interest.

On Mergable Coresets for Polytope Distance

We show that a constant-size constant-error coreset for polytope distance is simple to maintain under merges of coresets. However, increasing the size cannot improve the error bound significantly beyond that constant.

Sketching Multidimensional Time Series for Fast Discord Mining

Time series discords are a useful primitive for time series anomaly detection, and the matrix profile is capable of capturing discord effectively. There exist many research efforts to improve the scalability of discord discovery with respect to the length of time series. However, there is surprisingly little work focused on reducing the time complexity of matrix profile computation associated with dimensionality of a multidimensional time series. In this work, we propose a sketch for discord mining among multi-dimensional time series. After an initial pre-processing of the sketch as fast as reading the data, the discord mining has runtime independent of the dimensionality of the original data. On several real world examples from water treatment and transportation, the proposed algorithm improves the throughput by at least an order of magnitude (50X) and only has minimal impact on the quality of the approximated solution. Additionally, the proposed method can handle the dynamic addition or deletion of dimensions inconsequential overhead. This allows a data analyst to consider "what-if" scenarios in real time while exploring the data.