PosFormer: Recognizing Complex Handwritten Mathematical Expression with Position Forest Transformer

Handwritten Mathematical Expression Recognition (HMER) has wide applications in human-machine interaction scenarios, such as digitized education and automated offices. Recently, sequence-based models with encoder-decoder architectures have been commonly adopted to address this task by directly predicting LaTeX sequences of expression images. However, these methods only implicitly learn the syntax rules provided by LaTeX, which may fail to describe the position and hierarchical relationship between symbols due to complex structural relations and diverse handwriting styles. To overcome this challenge, we propose a position forest transformer (PosFormer) for HMER, which jointly optimizes two tasks: expression recognition and position recognition, to explicitly enable position-aware symbol feature representation learning. Specifically, we first design a position forest that models the mathematical expression as a forest structure and parses the relative position relationships between symbols. Without requiring extra annotations, each symbol is assigned a position identifier in the forest to denote its relative spatial position. Second, we propose an implicit attention correction module to accurately capture attention for HMER in the sequence-based decoder architecture. Extensive experiments validate the superiority of PosFormer, which consistently outperforms the state-of-the-art methods 2.03%/1.22%/2.00%, 1.83%, and 4.62% gains on the single-line CROHME 2014/2016/2019, multi-line M2E, and complex MNE datasets, respectively, with no additional latency or computational cost. Code is available at https://github.com/SJTU-DeepVisionLab/PosFormer.

New Distinguishers for Negation-Limited Weak Pseudorandom Functions

We show how to distinguish circuits with $\log k$ negations (a.k.a $k$-monotone functions) from uniformly random functions in $\exp\left(\tilde{O}\left(n^{1/3}k^{2/3}\right)\right)$ time using random samples. The previous best distinguisher, due to the learning algorithm by Blais, Cannone, Oliveira, Servedio, and Tan (RANDOM'15), requires $\exp\big(\tilde{O}(n^{1/2} k)\big)$ time. Our distinguishers are based on Fourier analysis on \emph{slices of the Boolean cube}. We show that some "middle" slices of negation-limited circuits have strong low-degree Fourier concentration and then we apply a variation of the classic Linial, Mansour, and Nisan "Low-Degree algorithm" (JACM'93) on slices. Our techniques also lead to a slightly improved weak learner for negation limited circuits under the uniform distribution.

Subspace Embedding and Linear Regression with Orlicz Norm

We consider a generalization of the classic linear regression problem to the case when the loss is an Orlicz norm. An Orlicz norm is parameterized by a non-negative convex function $G:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with $G(0)=0$: the Orlicz norm of a vector $x\in\mathbb{R}^n$ is defined as $ \|x\|_G=\inf\left\{\alpha>0\large\mid\sum_{i=1}^n G(|x_i|/\alpha)\leq 1\right\}. $ We consider the cases where the function $G(\cdot)$ grows subquadratically. Our main result is based on a new oblivious embedding which embeds the column space of a given matrix $A\in\mathbb{R}^{n\times d}$ with Orlicz norm into a lower dimensional space with $\ell_2$ norm. Specifically, we show how to efficiently find an embedding matrix $S\in\mathbb{R}^{m\times n},m<n$ such that $\forall x\in\mathbb{R}^{d},\Omega(1/(d\log n)) \cdot \|Ax\|_G\leq \|SAx\|_2\leq O(d^2\log n) \cdot \|Ax\|_G.$ By applying this subspace embedding technique, we show an approximation algorithm for the regression problem $\min_{x\in\mathbb{R}^d} \|Ax-b\|_G$, up to a $O(d\log^2 n)$ factor. As a further application of our techniques, we show how to also use them to improve on the algorithm for the $\ell_p$ low rank matrix approximation problem for $1\leq p<2$.