arXiv subjects

Guangjian Zhang

Publications and source records attributed to Guangjian Zhang.

4 recordsLinked to original sources

arXiv2026

Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. In vector-valued linear regression with square loss $\ell_{(x,y)}(W)=|Wx-y|_2^2$, where $x\in\mathbb{R}^d$, $y\in\mathbb{R}^m$ and the learner is the empirical risk minimizer of minimal Frobenius norm, we prove that the minimal budget of weighted examples that recovers the full-data loss on every finite dataset is exactly $n^*(d,m)=(m+1)d$. We further determine two more values of the weighted selection profile $F_w(d,m,n)$: at the near-threshold budget, $F_w(d,m,(m+1)d-1)=1+\frac{1}{dm^2}$, and at the spanning budget, $F_w(d,m,d)=d+1$ for every $m$, while $F_w(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_w(2,2,3)\in[13/8,15/8]$ and $F_w(2,2,4)\in[5/4,3/2]$, reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms, and establish strong structural evidence for the conjecture. The upper-bound techniques (a fixed-basis conic compression lemma, a determinant-facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems) are of independent interest. As a byproduct we correct an erroneous claim circulating in a recent unrefereed preprint, exhibiting an explicit dataset with $m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. All results are new only for $m\ge 2$; the scalar case $m=1$ is due to Hanneke et al.

cs.LG↗

arXiv2026

Exact Risk Ratios for Weighted Data Selection in Linear Regression

Hanneke, Moran, Shlimovich and Yehudayoff (COLT 2025) posed the following open problem. A selector sees a finite dataset $D \subseteq \mathbb{R}^d \times \mathbb{R}$, picks at most $n$ examples together with nonnegative weights, and hands the weighted least squares objective to the minimum-norm ERM. Writing $F_w(d,n)$ for the worst-case ratio between the loss of the returned predictor on all of $D$ and the optimal loss, they proved $F_w(d,n)=\infty$ for $n<d$, $F_w(d,d)=d+1$ and $F_w(d,n)=1$ for $n \ge 2d$, and asked for the value in the open regime $d<n<2d$. We determine this value in several cases. For every $d$ we prove $F_w(d,2d-1)=1+1/d$, which confirms a claim stated without proof in the original note. We further prove $F_w(3,4)=5/3$ and $F_w(4,5)=2$, the two smallest cells not covered by the endpoint formula. For every intermediate budget $n=d+k$ we prove the lower bound $F_w(d,d+k) \ge 1+Γ_{d,k}$, where $Γ_{d,k}$ is an explicit harmonic quantity over balanced partitions, and we show that this bound is the exact minimax value over the class of datasets whose whitened gradient systems carry an orthogonal circuit-block structure. All three exact values match $1+Γ_{d,k}$, and we conjecture that equality holds throughout the open regime. The upper bound proofs run on a common geometric spine: a rigidity theorem for positive spanning configurations of loss gradients, classifications and structural reductions of small positive bases in $\mathbb{R}^3$ and $\mathbb{R}^4$, and a dimension-free extremal-basis argument that converts sign-cone geometry into five-point selections. We also give explicit counterexamples showing that several shorter routes fail, and constructive polynomial-time selection algorithms for all proved cases.

cs.LG↗

arXiv2023

Exploiting Point-Wise Attention in 6D Object Pose Estimation Based on Bidirectional Prediction

Traditional geometric registration based estimation methods only exploit the CAD model implicitly, which leads to their dependence on observation quality and deficiency to occlusion. To address the problem,the paper proposes a bidirectional correspondence prediction network with a point-wise attention-aware mechanism. This network not only requires the model points to predict the correspondence but also explicitly models the geometric similarities between observations and the model prior. Our key insight is that the correlations between each model point and scene point provide essential information for learning point-pair matches. To further tackle the correlation noises brought by feature distribution divergence, we design a simple but effective pseudo-siamese network to improve feature homogeneity. Experimental results on the public datasets of LineMOD, YCB-Video, and Occ-LineMOD show that the proposed method achieves better performance than other state-of-the-art methods under the same evaluation criteria. Its robustness in estimating poses is greatly improved, especially in an environment with severe occlusions.

cs.CV↗

arXiv2017

On subcritically Stein fillable 5-manifolds

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the 5-sphere the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

math.GT↗