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 study and provide instance-optimal algorithms in differential privacy by extending and approximating the inverse sensitivity mechanism.

Earlier schemes could only handle stochastic substitutions anddeletions, andthus weareaiming foramore natural and appealing robustness guarantee that holds with respect to editdistance.

Everybody is taking a stab at designing artificial intelligence processors, or electronic chips that could become the brains of computers that act as if they were humans. The latest to tackle the task of designing AI chips is Vinod Dham, a former Intel executive known as the "father of the Pentium." He has teamed up with some younger chip designers to build RAP chips, or real AI processors. At AlphaICs, the team is creating a coprocessor chip that can do agent-based artificial intelligence. These RAP chips could one day be deployed in computing devices and autonomous cars to make decisions at lightning speeds, or in data centers on a massive scale.