Hengzhi He
Publications
Tensor-normal maximum likelihood estimation at the operator-norm sample threshold
Let $X_1,\ldots,X_n$ be independent Gaussian tensors in $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_k}$ whose covariance is a Kronecker product of $k$ unknown positive-definite factors, and put $D=\prod_{a=1}^k d_a$ and $d_{\max}=\max_a d_a$. A recent result of Franks et al. (2026) established condition-number-free nonasymptotic guarantees for the tensor-normal maximum likelihood estimator under the sample threshold $nD\gtrsim k^2 d_{\max}^3$. They asked whether the cubic dependence on $d_{\max}$ could be replaced by the operator-norm scale $d_{\max}^2$. We answer this question affirmatively. We prove that, for $t\geq 1$, the maximum likelihood estimator exists uniquely with high probability whenever $nD\geq Ck^2 d_{\max}^2 t^2$, and satisfies $d_{\mathrm{FR}}(\widehatΘ,Θ)\leq Ct\sqrt{k}\,d_{\max}/\sqrt{n}$ and $d_{\mathrm{FR}}(\widehatΘ_a,Θ_a)\leq Ct\sqrt{k d_a}\,d_{\max}/\sqrt{nD}$. For every mode of largest dimension, we also obtain the sharp Thompson bound $d_{\mathrm{op}}(\widehatΘ_a,Θ_a)\leq Ct\,d_{\max}/\sqrt{nD}$. No sparsity, condition-number bound or warm start is assumed. For fixed $k$, the threshold has the information-theoretically optimal dependence on $d_{\max}$, and the displayed rates for the full precision and the largest factor match Gaussian minimax lower bounds up to a factor $\sqrt{k}$. The proof extends a random Gram bound for local group-orbit directions to the full local Lie algebra, transports it to a fixed Thompson ball by exact conjugation, and combines sensitivity of a constrained maximum likelihood estimator with an equivariant Kirszbraun extension and Gaussian concentration. This removes the Frobenius-to-operator loss responsible for the previous extra factor $d_{\max}$ and resolves the explicit open problem posed in the earlier work.
Enhancing TableQA through Verifiable Reasoning Trace Reward
A major challenge in training TableQA agents, compared to standard text- and image-based agents, is that answers cannot be inferred from a static input but must be reasoned through stepwise transformations of the table state, introducing multi-step reasoning complexity and environmental interaction. This leads to a research question: Can explicit feedback on table transformation action improve model reasoning capability? In this work, we introduce RE-Tab, a plug-and-play framework that architecturally enhances trajectory search via lightweight, training-free reward modeling by formulating the problem as a Partially Observable Markov Decision Process. We demonstrate that providing explicit verifiable rewards during State Transition (``What is the best action?'') and Simulative Reasoning (``Am I sure about the output?'') is crucial to steer the agent's navigation in table states. By enforcing stepwise reasoning with reward feedback in table transformations, RE-Tab achieves state-of-the-art performance in TableQA with almost 25\% drop in inference cost. Furthermore, a direct plug-and-play implementation of RE-Tab brings up to 41.77% improvement in QA accuracy and 33.33% drop in test-time inference samples for consistent answer. Consistent improvement pattern across various LLMs and state-of-the-art benchmarks further confirms RE-Tab's generalisability. The repository is available at https://github.com/ThomasK1018/RE_Tab .