We introduce the zip tree, a form of randomized binary search tree that integrates previous ideas into one practical, performant, and pleasant-to-implement package. A zip tree is a binary search tree in which each node has a numeric rank and the tree is (max)-heap-ordered with respect to ranks, with rank ties broken in favor of smaller keys. Zip trees are essentially treaps (Seidel and Aragon 1996), except that ranks are drawn from a geometric distribution instead of a uniform distribution, and we allow rank ties. These changes enable us to use fewer random bits per node. We perform insertions and deletions by unmerging and merging paths ("unzipping" and "zipping") rather than by doing rotations, which avoids some pointer changes and improves efficiency. The methods of zipping and unzipping take inspiration from previous top-down approaches to insertion and deletion (Stephenson 1980; Mart\'inez and Roura 1998; Sprugnoli 1980). From a theoretical standpoint, this work provides two main results. First, zip trees require only $O(\log \log n)$ bits (with high probability) to represent the largest rank in an $n$-node binary search tree; previous data structures require $O(\log n)$ bits for the largest rank. Second, zip trees are naturally isomorphic to skip lists (Pugh 1990), and simplify the mapping of (Dean and Jones 2007) between skip lists and binary search trees.
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We study complexity classes of local problems on regular trees from the perspective of distributed local algorithms and descriptive combinatorics. We show that, surprisingly, some deterministic local complexity classes from the hierarchy of distributed computing exactly coincide with well studied classes of problems in descriptive combinatorics. Namely, we show that a local problem admits a continuous solution if and only if it admits a local algorithm with local complexity $O(\log^* n)$, and a Baire measurable solution if and only if it admits a local algorithm with local complexity $O(\log n)$.
Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this paper we address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples -- taken as a mean persistence measure computed from the subsamples -- is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.
Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.
Modern web services routinely provide REST APIs for clients to access their functionality. These APIs present unique challenges and opportunities for automated testing, driving the recent development of many techniques and tools that generate test cases for API endpoints using various strategies. Understanding how these techniques compare to one another is difficult, as they have been evaluated on different benchmarks and using different metrics. To fill this gap, we performed an empirical study aimed to understand the landscape in automated testing of REST APIs and guide future research in this area. We first identified, through a systematic selection process, a set of 10 state-of-the-art REST API testing tools that included tools developed by both researchers and practitioners. We then applied these tools to a benchmark of 20 real-world open-source RESTful services and analyzed their performance in terms of code coverage achieved and unique failures triggered. This analysis allowed us to identify strengths, weaknesses, and limitations of the tools considered and of their underlying strategies, as well as implications of our findings for future research in this area.
Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.
We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all $k$-vertex subgraphs of an $n$-vertex graph. When $k$ is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is $k$, the number of "infected" vertices in the network. For a certain range of parameters the running time of our algorithms on $n$-vertex graphs is $2^{o(k)}\textrm{poly}(n)$, improving upon the $2^{\Omega(k)}\textrm{poly}(n)$ performance of the best-known algorithms designed for worst-case instances of these edge deletion problems.
Imposing consistency through proxy tasks has been shown to enhance data-driven learning and enable self-supervision in various tasks. This paper introduces novel and effective consistency strategies for optical flow estimation, a problem where labels from real-world data are very challenging to derive. More specifically, we propose occlusion consistency and zero forcing in the forms of self-supervised learning and transformation consistency in the form of semi-supervised learning. We apply these consistency techniques in a way that the network model learns to describe pixel-level motions better while requiring no additional annotations. We demonstrate that our consistency strategies applied to a strong baseline network model using the original datasets and labels provide further improvements, attaining the state-of-the-art results on the KITTI-2015 scene flow benchmark in the non-stereo category. Our method achieves the best foreground accuracy (4.33% in Fl-all) over both the stereo and non-stereo categories, even though using only monocular image inputs.
We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 distance in $O(n\cdot \text{poly}(1/\epsilon))$ time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of $n$, but exponential in $1/\epsilon$. We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian $A$, this moment matching method returns an $\epsilon$ approximation to the spectral density using just $O({1}/{\epsilon})$ matrix-vector products with $A$. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used \emph{kernel polynomial method} (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.
Decomposition-based evolutionary algorithms have become fairly popular for many-objective optimization in recent years. However, the existing decomposition methods still are quite sensitive to the various shapes of frontiers of many-objective optimization problems (MaOPs). On the one hand, the cone decomposition methods such as the penalty-based boundary intersection (PBI) are incapable of acquiring uniform frontiers for MaOPs with very convex frontiers. On the other hand, the parallel reference lines of the parallel decomposition methods including the normal boundary intersection (NBI) might result in poor diversity because of under-sampling near the boundaries for MaOPs with concave frontiers. In this paper, a collaborative decomposition method is first proposed to integrate the advantages of parallel decomposition and cone decomposition to overcome their respective disadvantages. This method inherits the NBI-style Tchebycheff function as a convergence measure to heighten the convergence and uniformity of distribution of the PBI method. Moreover, this method also adaptively tunes the extent of rotating an NBI reference line towards a PBI reference line for every subproblem to enhance the diversity of distribution of the NBI method. Furthermore, a collaborative decomposition-based evolutionary algorithm (CoDEA) is presented for many-objective optimization. A collaborative decomposition-based environmental selection mechanism is primarily designed in CoDEA to rank all the individuals associated with the same PBI reference line in the boundary layer and pick out the best ranks. CoDEA is compared with several popular algorithms on 85 benchmark test instances. The experimental results show that CoDEA achieves high competitiveness benefiting from the collaborative decomposition maintaining a good balance among the convergence, uniformity, and diversity of distribution.
In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.
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