This paper studies \emph{linear} and \emph{affine} error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to have information rate at most $1/2$ (achieved by the trivial 2-fold repetition code). Previously, it was (erroneously) reported that more generally no non-trivial linear codes correcting $k$ deletions exist, i.e., that the $(k+1)$-fold repetition codes and its rate of $1/(k+1)$ are basically optimal for any $k$. We disprove this and show the existence of binary linear codes of length $n$ and rate just below $1/2$ capable of correcting $\Omega(n)$ insertions and deletions. This identifies rate $1/2$ as a sharp threshold for recovery from deletions for linear codes, and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically-good linear code for Hamming errors into an asymptotically-good linear code for insdel errors. Lastly, we show that the $\frac{1}{2}$-rate limitation does not hold for affine codes by giving an explicit affine code of rate $1-\epsilon$ which can efficiently correct a constant fraction of insdel errors.
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Recent studies have shown that autoencoder-based models can achieve superior performance on anomaly detection tasks due to their excellent ability to fit complex data in an unsupervised manner. In this work, we propose a novel autoencoder-based model, named StackVAE-G that can significantly bring the efficiency and interpretability to multivariate time series anomaly detection. Specifically, we utilize the similarities across the time series channels by the stacking block-wise reconstruction with a weight-sharing scheme to reduce the size of learned models and also relieve the overfitting to unknown noises in the training data. We also leverage a graph learning module to learn a sparse adjacency matrix to explicitly capture the stable interrelation structure among multiple time series channels for the interpretable pattern reconstruction of interrelated channels. Combining these two modules, we introduce the stacking block-wise VAE (variational autoencoder) with GNN (graph neural network) model for multivariate time series anomaly detection. We conduct extensive experiments on three commonly used public datasets, showing that our model achieves comparable (even better) performance with the state-of-the-art modelsand meanwhile requires much less computation and memory cost. Furthermore, we demonstrate that the adjacency matrix learned by our model accurately captures the interrelation among multiple channels, and can provide valuable information for failure diagnosis applications.
Locally Decodable Codes (LDCs) are error-correcting codes $C:\Sigma^n\rightarrow \Sigma^m$ with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson et al. showed how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Insdel LDCs were first studied by Ostrovsky and Paskin-Cherniavsky, and are further motivated by recent advances in DNA random access bio-technologies, in which the goal is to retrieve individual files from a DNA storage database. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries, over the binary alphabet. This answers a question explicitly raised by Gur and Lachish. Our result exhibits a "phase-transition"-type behavior on the codeword length for constant-query Hamming RLDCs. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with with parameters matching those of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.
In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution effect if, as $k\to \infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter 2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania, Sauter 2013] showed that the $hp$-FEM (where accuracy is increased by decreasing the meshwidth $h$ and increasing the polynomial degree $p$) applied to a variety of constant-coefficient Helmholtz problems does not suffer from the pollution effect. The heart of the proofs of these results is a PDE result splitting the solution of the Helmholtz equation into "high" and "low" frequency components. In this expository paper we prove this splitting for the constant-coefficient Helmholtz equation in full space (i.e., in $\mathbb{R}^d$) using only integration by parts and elementary properties of the Fourier transform; this is in contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses somewhat-involved bounds on Bessel and Hankel functions. The proof in this paper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2022] of this splitting for the variable-coefficient Helmholtz equation in full space; indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticated tools that reduce to the elementary ones above for constant coefficients.
Locally Decodable Codes (LDCs) are error-correcting codes for which individual message symbols can be quickly recovered despite errors in the codeword. LDCs for Hamming errors have been studied extensively in the past few decades, where a major goal is to understand the amount of redundancy that is necessary and sufficient to decode from large amounts of error, with small query complexity. In this work, we study LDCs for insertion and deletion errors, called Insdel LDCs. Their study was initiated by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), who gave a reduction from Hamming LDCs to Insdel LDCs with a small blowup in the code parameters. On the other hand, the only known lower bounds for Insdel LDCs come from those for Hamming LDCs, thus there is no separation between them. Here we prove new, strong lower bounds for the existence of Insdel LDCs. In particular, we show that $2$-query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all $q$-query Insdel LDCs with constant $q$. For $q \ge 3$ our bounds are exponential in the existing lower bounds for Hamming LDCs. Furthermore, our exponential lower bounds continue to hold for adaptive decoders, and even in private-key settings where the encoder and decoder share secret randomness. This exhibits a strict separation between Hamming LDCs and Insdel LDCs. Our strong lower bounds also hold for the related notion of Insdel LCCs (except in the private-key setting), due to an analogue to the Insdel notions of a reduction from Hamming LCCs to LDCs. Our techniques are based on a delicate design and analysis of hard distributions of insertion and deletion errors, which depart significantly from typical techniques used in analyzing Hamming LDCs.
Prophet inequalities for rewards maximization are fundamental results from optimal stopping theory with several applications to mechanism design and online optimization. We study the cost minimization counterpart of the classical prophet inequality, where one is facing a sequence of costs $X_1, X_2, \dots, X_n$ in an online manner and must ''stop'' at some point and take the last cost seen. Given that the $X_i$'s are independent, drawn from known distributions, the goal is to devise a stopping strategy $S$ (online algorithm) that minimizes the expected cost. We first observe that if the $X_i$'s are not identically distributed, then no strategy can achieve a bounded approximation, no matter if the arrival order is adversarial or random. This leads us to consider the case where the $X_i$'s are I.I.D.. For the I.I.D. case, we give a complete characterization of the optimal stopping strategy. We show that it achieves a (distribution-dependent) constant-factor approximation to the prophet's cost for almost all distributions and that this constant is tight. In particular, for distributions for which the integral of the hazard rate is a polynomial $H(x) = \sum_{i=1}^k a_i x^{d_i}$, where $d_1 < \dots < d_k$, the approximation factor is $\lambda(d_1)$, a decreasing function of $d_1$. Furthermore, for MHR distributions, we show that this constant is at most $2$, and this is again tight. We also analyze single-threshold strategies for the cost prophet inequality problem. We design a threshold that achieves a $\operatorname{O}(\operatorname{polylog}n)$-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound. We believe that our results are of independent interest for analyzing approximately optimal (posted price-style) mechanisms for procuring items.
In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes $\mu$, the strategy is stable both in terms of mean-square error and average bit rate: by reducing $\mu$, it is possible to keep the estimation errors small (on the order of $\mu$) without increasing indefinitely the bit rate as $\mu\rightarrow 0$. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other graph sparsifiers using linear sketching, and obtained near-linear upper bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21]. All these linear sketching algorithms, however, only work on unweighted graphs. In this paper, we initiate the study of weighted graph sparsification by linear sketching by investigating a natural class of linear sketches that we call incidence sketches, in which each measurement is a linear combination of the weights of edges incident on a single vertex. Our results are: 1. Weighted cut sparsification: We give an algorithm that computes a $(1 + \epsilon)$-cut sparsifier using $\tilde{O}(n \epsilon^{-3})$ linear measurements, which is nearly optimal. 2. Weighted spectral sparsification: We give an algorithm that computes a $(1 + \epsilon)$-spectral sparsifier using $\tilde{O}(n^{6/5} \epsilon^{-4})$ linear measurements. Complementing our algorithm, we then prove a superlinear lower bound of $\Omega(n^{21/20-o(1)})$ measurements for computing some $O(1)$-spectral sparsifier using incidence sketches. 3. Weighted spanner computation: We focus on graphs whose largest/smallest edge weights differ by an $O(1)$ factor, and prove that, for incidence sketches, the upper bounds obtained by~[Filtser-Kapralov-Nouri, SODA'21] are optimal up to an $n^{o(1)}$ factor.
Optical coherence tomography (OCT) is a micrometer-scale, volumetric imaging modality that has become a clinical standard in ophthalmology. OCT instruments image by raster-scanning a focused light spot across the retina, acquiring sequential cross-sectional images to generate volumetric data. Patient eye motion during the acquisition poses unique challenges: Non-rigid, discontinuous distortions can occur, leading to gaps in data and distorted topographic measurements. We present a new distortion model and a corresponding fully-automatic, reference-free optimization strategy for computational motion correction in orthogonally raster-scanned, retinal OCT volumes. Using a novel, domain-specific spatiotemporal parametrization of forward-warping displacements, eye motion can be corrected continuously for the first time. Parameter estimation with temporal regularization improves robustness and accuracy over previous spatial approaches. We correct each A-scan individually in 3D in a single mapping, including repeated acquisitions used in OCT angiography protocols. Specialized 3D forward image warping reduces median runtime to < 9 s, fast enough for clinical use. We present a quantitative evaluation on 18 subjects with ocular pathology and demonstrate accurate correction during microsaccades. Transverse correction is limited only by ocular tremor, whereas submicron repeatability is achieved axially (0.51 um median of medians), representing a dramatic improvement over previous work. This allows assessing longitudinal changes in focal retinal pathologies as a marker of disease progression or treatment response, and promises to enable multiple new capabilities such as supersampled/super-resolution volume reconstruction and analysis of pathological eye motion occuring in neurological diseases.
Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes known as characteristic components.We propose an alternative coarse-grid that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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