In theory, the techniques can speed up matrix operations, however, in practice, they are not fast enough.

We believe there is a misconception here.

In this paper, we propose and investigate algorithms for the structured orthogonal dictionary learning problem. First, we investigate the case when the dictionary is a Householder matrix. We give sample complexity results and show theoretically guaranteed approximate recovery (in the $l_{\infty}$ sense) with optimal computational complexity. We then attempt to generalize these techniques when the dictionary is a product of a few Householder matrices. We numerically validate these techniques in the sample-limited setting to show performance similar to or better than existing techniques while having much improved computational complexity.

Motivated by orthogonal dictionary learning problems, we propose a novel method for matrix factorization, where the data matrix $\mathbf{Y}$ is a product of a Householder matrix $\mathbf{H}$ and a binary matrix $\mathbf{X}$. First, we show that the exact recovery of the factors $\mathbf{H}$ and $\mathbf{X}$ from $\mathbf{Y}$ is guaranteed with $\Omega(1)$ columns in $\mathbf{Y}$ . Next, we show approximate recovery (in the $l\infty$ sense) can be done in polynomial time($O(np)$) with $\Omega(\log n)$ columns in $\mathbf{Y}$ . We hope the techniques in this work help in developing alternate algorithms for orthogonal dictionary learning.

Bilingual lexicon induction induces the word translations by aligning independently trained word embeddings in two languages. Existing approaches generally focus on minimizing the distances between words in the aligned pairs, while suffering from low discriminative capability to distinguish the relative orders between positive and negative candidates. In addition, the mapping function is globally shared by all words, whose performance might be hindered by the deviations in the distributions of different languages. In this work, we propose a novel ranking-oriented induction model RAPO to learn personalized mapping function for each word. RAPO is capable of enjoying the merits from the unique characteristics of a single word and the cross-language isomorphism simultaneously. Extensive experimental results on public datasets including both rich-resource and low-resource languages demonstrate the superiority of our proposal. Our code is publicly available in \url{https://github.com/Jlfj345wf/RAPO}.

Various Neural Networks employ time-consuming matrix operations like matrix inversion. Many such matrix operations are faster to compute given the Singular Value Decomposition (SVD). Previous work allows using the SVD in Neural Networks without computing it. In theory, the techniques can speed up matrix operations, however, in practice, they are not fast enough. We present an algorithm that is fast enough to speed up several matrix operations. The algorithm increases the degree of parallelism of an underlying matrix multiplication $H\cdot X$ where $H$ is an orthogonal matrix represented by a product of Householder matrices. Code is available at www.github.com/AlexanderMath/fasth .

Decision trees are a popular technique in statistical data classification. They recursively partition the feature space into disjoint sub-regions until each sub-region becomes homogeneous with respect to a particular class. The basic Classification and Regression Tree (CART) algorithm partitions the feature space using axis parallel splits. When the true decision boundaries are not aligned with the feature axes, this approach can produce a complicated boundary structure. Oblique decision trees use oblique decision boundaries to potentially simplify the boundary structure. The major limitation of this approach is that the tree induction algorithm is computationally expensive. In this article we present a new decision tree algorithm, called HHCART. The method utilizes a series of Householder matrices to reflect the training data at each node during the tree construction. Each reflection is based on the directions of the eigenvectors from each classes' covariance matrix. Considering axis parallel splits in the reflected training data provides an efficient way of finding oblique splits in the unreflected training data. Experimental results show that the accuracy and size of the HHCART trees are comparable with some benchmark methods in the literature. The appealing feature of HHCART is that it can handle both qualitative and quantitative features in the same oblique split.