The interplay of different metrics for the construction of constant dimension codes

<p style='text-indent:20px;'>A basic problem for constant dimension codes is to determine the maximum possible size <inline-formula><tex-math id="M1">\begin{document}$ A_q(n,d;k) $\end{document}</tex-math></inline-formula> of a set of <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-dimensional subspaces in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_q^n $\end{document}</tex-math></inline-formula>, called codewords, such that the subspace distance satisfies <inline-formula><tex-math id="M4">\begin{document}$ d_S(U,W): = 2k-2\dim(U\cap W)\ge d $\end{document}</tex-math></inline-formula> for all pairs of different codewords <inline-formula><tex-math id="M5">\begin{document}$ U $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for <inline-formula><tex-math id="M7">\begin{document}$ A_q(n,d;k) $\end{document}</tex-math></inline-formula> are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show the potential for further improvements. As examples we give improved constructions for the cases <inline-formula><tex-math id="M8">\begin{document}$ A_q(10,4;5) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ A_q(11,4;4) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ A_q(12,6;6) $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M11">\begin{document}$ A_q(15,4;4) $\end{document}</tex-math></inline-formula>. We also derive general upper bounds for subcodes arising in those constructions.</p>

On sets of subspaces with two intersection dimensions and a geometrical junta bound

In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a junta (Combin Probab Comput 18(1–2):107–122, 2009); i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.

A Proposal-Based Approach for Activity Image-to-Video Retrieval

Activity image-to-video retrieval task aims to retrieve videos containing the similar activity as the query image, which is a challenging task because videos generally have many background segments irrelevant to the activity. In this paper, we utilize R-C3D model to represent a video by a bag of activity proposals, which can filter out background segments to some extent. However, there are still noisy proposals in each bag. Thus, we propose an Activity Proposal-based Image-to-Video Retrieval (APIVR) approach, which incorporates multi-instance learning into cross-modal retrieval framework to address the proposal noise issue. Specifically, we propose a Graph Multi-Instance Learning (GMIL) module with graph convolutional layer, and integrate this module with classification loss, adversarial loss, and triplet loss in our cross-modal retrieval framework. Moreover, we propose geometry-aware triplet loss based on point-to-subspace distance to preserve the structural information of activity proposals. Extensive experiments on three widely-used datasets verify the effectiveness of our approach.

Bounds on Subspace Codes Based on Orthogonal Space Over Finite Fields of Characteristic 2

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length [Formula: see text], size [Formula: see text], minimum subspace distance [Formula: see text] based on [Formula: see text]-dimensional totally singular subspace in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over finite fields [Formula: see text] of characteristic 2, denoted by [Formula: see text], are presented, where [Formula: see text] is a positive integer, [Formula: see text], [Formula: see text], [Formula: see text]. Then, we prove that [Formula: see text] codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text], where [Formula: see text] denotes the collection of all the [Formula: see text]-dimensional totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code [Formula: see text] in [Formula: see text] are provided, where [Formula: see text] denotes the collection of all the totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2.

Principal Component Analysis in the Local Differential Privacy Model

In this paper, we study the Principal Component Analysis (PCA) problem under the (distributed) non-interactive local differential privacy model. For the low dimensional case, we show the optimal rate for the private minimax risk of the k-dimensional PCA using the squared subspace distance as the measurement. For the high dimensional row sparse case, we first give a lower bound on the private minimax risk, . Then we provide an efficient algorithm to achieve a near optimal upper bound. Experiments on both synthetic and real world datasets confirm the theoretical guarantees of our algorithms.