For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ \gamma_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = \Omega\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.
Honghao Lin, V. Mirrokni, David P. Woodruff· 0 citations
We introduce TCS-Bench, a benchmark for evaluating Large Language Models (LLMs) on research-level Theoretical Computer Science (TCS) proof generation. TCS-Bench consists of theorem-proving tasks from papers published at top theoretical computer science venues (STOC, FOCS, and SODA). Each task provides the necessary context to derive a self-contained proof for a target result. We evaluate state-of-the-art models on this benchmark. We verify the correctness of generated proofs via a verification agent, and further benchmark the verifier against human-expert proof judgements on a set of target statements and generated proofs pairs. Our reference verifier achieves over 90% accuracy on the expert labeled set.
Vincent Cohen-Addad, Dimitris Paparas, E. V. Wijland et al.· 1 citation
The quadratic cost of attention-based sequence models for long contexts has motivated a growing line of research on memory-based models that can compress context into a compact state. However, most existing memory models expose a static memory throughout the entire sequence. Because early tokens face no compression pressure, they occupy too many degrees of freedom and"pollute"the memory state, leaving little capacity for later context and increasing interference between what is stored and what arrives next. We study a new paradigm of incremental memory activation, where the effective capacity of memory is progressively expanded as the context grows. Imposing an early bottleneck forces the model to compress history more effectively, while unlocking fresh capacity over time reduces interference and improves retention of later context. We instantiate this paradigm in Proteus, a straightforward mechanism that can be incorporated into a broad class of neural memory architectures at no additional cost. We apply Proteus to state-of-the-art models, including SWLA, Comba, Titans, and Hope-Attention, and observe consistent improvements on standard language modeling and reasoning, as well as on long-context retrieval and understanding, with gains that grow at longer context lengths. Overall, our results show that static memory is suboptimal and that scheduling effective capacity is a simple and broadly applicable tool for sequence modeling.
Reza Bayat, Ali Behrouz, V. Mirrokni et al.· 0 citations
Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3\pi\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.
Honghao Lin, V. Mirrokni, David P. Woodruff· 0 citations
Experiments show that HVA establishes a new Pareto frontier for training-free sparse attention in video diffusion, reducing end-to-end latency by up to $2.13\times while improving fidelity over existing training-free sparse attention baselines.
Dongyeun Lee, A. Zandieh, V. Mirrokni et al.· 1 citation
The Paper Assistant Tool is introduced, an agentic AI framework built for deep scientific review and verification and able to identify deeper issues than a single model call alone, achieving a 34% improvement over zero-shot recall on mathematical errors in the SPOT benchmark.
Rajesh Jayaram, Drew Tyler, David P. Woodruff et al.· arXiv.org· 0 citations