我们考虑的问题，找到一个低的差异着色的稀疏集系统中的每个元素位于最多$t$集。我们给出了一个算法，找到一个着色与差异$O((t \log n \log s)^{1/2})$，其中$s$是一个集的最大基数。这改进了基于部分着色方法的算法变体的$O(t^{1/2} \log n)$的先前构造性界限，并且对于小$s$（例如$s=\textrm{poly}(t)$），由于Banaszczyk而接近非构造性$O((t \log n)^{1/2})$界限。以前，即使对于$s = O(t^2)$，也没有已知比$O(t^{1/2}\log n)$更好的算法结果。我们的方法是相当强大的，我们给出了一些改进和扩展。例如，我们得到的着色满足更强的大小敏感性，在集合系统中的每个集合$S$招致$O((t \log n \log |S|)^{1/2})$的差异。另一个变体可以用来基本上匹配Banaszczyk的边界，即使$s$是任意大的实例。最后，这些结果也直接扩展到更一般的Komlós设置。
英文摘要：
We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most $t$ sets. We give an algorithm that finds a coloring with discrepancy $O((t \log n \log s)^{1/2})$ where $s$ is the maximum cardinality of a set. This improves upon the previous constructive bound of $O(t^{1/2} \log n)$ based on algorithmic variants of the partial coloring method, and for small $s$ (e.g.$s=\textrm{poly}(t)$) comes close to the non-constructive $O((t \log n)^{1/2})$ bound due to Banaszczyk. Previously, no algorithmic results better than $O(t^{1/2}\log n)$ were known even for $s = O(t^2)$. Our method is quite robust and we give several refinements and extensions. For example, the coloring we obtain satisfies the stronger size-sensitive property that each set $S$ in the set system incurs an $O((t \log n \log |S|)^{1/2})$ discrepancy. Another variant can be used to essentially match Banaszczyk's bound for a wide class of instances even where $s$ is arbitrarily large. Finally, these results also extend directly to the more general Komlós setting.
This report will be a literature review on a result in algorithmic discrepancy theory. We will begin by providing a quick overview on discrepancy theory and some major results in the field, and then focus on an important result by Shachar Lovett and Raghu Meka. We restate the main algorithm and ideas of the paper, and rewrite proofs for some of the major results in the paper.
Stochastic scientific models and machine learning optimization estimators have a large number of variables; hence computing large sparse Jacobians and Hessians is important. Algorithmic differentiation (AD) greatly reduces the programming effort required to obtain the sparsity patterns and values for these matrices. We present forward, reverse, and subgraph methods for computing sparse Jacobians and Hessians. Special attention is given the the subgraph method because it is new. The coloring and compression steps are not necessary when computing sparse Jacobians and Hessians using subgraphs. Co…
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $Δ$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le Δ^\frac{2\varepsilon}{1+2\varepsilon}/(\logΔ)^2$, with $\lfloor(1+\varepsilon)Δ/\log(Δ/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) an…
Efficiently computing low discrepancy colorings of various set systems, has been studied extensively since the breakthrough work by Bansal (FOCS 2010), who gave the first polynomial time algorithms for several important settings, including for general set systems, sparse set systems and for set systems with bounded hereditary discrepancy. The hereditary discrepancy of a set system, is the maximum discrepancy over all set systems obtainable by deleting a subset of the ground elements. While being polynomial time, Bansal's algorithms were not practical, with e.g. his algorithm for the hereditary…
In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix $A \in \mathbb{R}^{m \times n}$ in terms of the maximum $|\det(B)|^{1/k}$ over all $k \times k$ submatrices $B$ of $A$. We show algorithmically that this determinant lower bound can be off by at most a factor of $O(\sqrt{\log (m) \cdot \log (n)})$, improving over the previous bound of $O(\log(mn) \cdot \sqrt{\log (n)})$ given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies $\mathrm{herdisc}(\mathcal{F}_1 \cup \mathcal{F}_2) \leq O(\…
ETH Zürich（苏黎世联邦理工学院）；Carnegie-Mellon University（卡内基梅隆大学）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
We study efficient algorithms for Sparse PCA in standard statistical models (spiked covariance in its Wishart form). Our goal is to achieve optimal recovery guarantees while being resilient to small perturbations. Despite a long history of prior works, including explicit studies of perturbation resilience, the best known algorithmic guarantees for Sparse PCA are fragile and break down under small adversarial perturbations. We observe a basic connection between perturbation resilience and \emph{certifying algorithms} that are based on certificates of upper bounds on sparse eigenvalues of random…
