We develop a geometric framework that links objective accuracy to structural recovery in prototype-based clustering. The analysis is algorithm-agnostic and applies to a broad class of admissible loss functions. We define a clustering condition number that compares within-cluster scale to the minimum loss increase required to move a point across a cluster boundary. When this quantity is small, any solution with a small suboptimality gap must also have a small misclassification error relative to a benchmark partition. The framework also clarifies a fundamental trade-off between robustness and sensitivity to cluster imbalance, leading to sharp phase transitions for exact recovery under different objectives. The guarantees are deterministic and non-asymptotic, and they separate the role of algorithmic accuracy from the intrinsic geometric difficulty of the instance. We further show that errors concentrate near cluster boundaries and that sufficiently deep cluster cores are recovered exactly under strengthened local margins. Together, these results provide a geometric principle for interpreting low objective values as reliable evidence of meaningful clustering structure.

We are interested in a framework of online learning with kernels for lowdimensional, but large-scale and potentially adversarial datasets. We study the computational and theoretical performance of online variations of kernel Ridge regression.

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We further show that optimistic posterior sampling can control this Hellinger distance, when we measure model error via data likelihood. This technique allows us to design and analyze unified posterior sampling algorithms with state-of-the-art sample complexity guarantees for many model-based RL settings.

We study pure exploration in bandits, where the dimension of the feature representation can be much larger than the number of arms.

Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which canbeimplemented efficiently andwhich perform wellinnumericalexperiments.

Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which canbeimplemented efficiently andwhich perform wellinnumericalexperiments.

Existing visual token pruning methods target prompt alignment and visual preservation with static strategies, overlooking the varying relative importance of these objectives across tasks, which leads to inconsistent performance. To address this, we derive the first closed-form error bound for visual token pruning based on the Hausdorff distance, uniformly characterizing the contributions of both objectives. Moreover, leveraging $ε$-covering theory, we reveal an intrinsic trade-off between these objectives and quantify their optimal attainment levels under a fixed budget. To practically handle this trade-off, we propose Multi-Objective Balanced Covering (MoB), which reformulates visual token pruning as a bi-objective covering problem. In this framework, the attainment trade-off reduces to budget allocation via greedy radius trading. MoB offers a provable performance bound and linear scalability with respect to the number of input visual tokens, enabling adaptation to challenging pruning scenarios. Extensive experiments show that MoB preserves 96.4% of performance for LLaVA-1.5-7B using only 11.1% of the original visual tokens and accelerates LLaVA-Next-7B by 1.3-1.5$\times$ with negligible performance loss. Additionally, evaluations on Qwen2-VL and Video-LLaVA confirm that MoB integrates seamlessly into advanced MLLMs and diverse vision-language tasks.