Burak Bartan

Bio: Burak Bartan is an academic researcher from Stanford University. The author has contributed to research in topics: Optimization problem & Convex optimization. The author has an hindex of 7, co-authored 24 publications receiving 169 citations.

Repairing Multiple Failures for Scalar MDS Codes

Abstract: In distributed storage, erasure codes (like Reed–Solomon Codes) are often employed to provide reliability. In this setting, it is desirable to be able to repair one or more failed nodes while minimizing the repair bandwidth . In this paper, motivated by Reed-Solomon codes, we study the problem of repairing multiple failed nodes in a scalar MDS code. We extend the framework of (Guruswami and Wootters, 2017) to give a framework for constructing repair schemes for multiple failures in general scalar MDS codes in the centralized repair model. We then specialize our framework to Reed–Solomon codes, and also extend and improve upon recent results of (Dau et al. , 2017).

Repairing multiple failures for scalar MDS codes

Abstract: In distributed storage, erasure codes — like Reed-Solomon Codes — are often employed to provide reliability. In this setting, it is desirable to be able to repair one or more failed nodes while minimizing the repair bandwidth. In this work, motivated by Reed-Solomon codes, we study the problem of repairing multiple failed nodes in a scalar MDS code. We extend the framework of (Guruswami and Wootters, 2017) to give a framework for constructing repair schemes for multiple failures in general scalar MDS codes, in the centralized repair model. We then specialize our framework to Reed-Solomon codes, and extend and improve upon recent results of (Dau et al., 2017).

Sparse representation of two- and three-dimensional images with fractional Fourier, Hartley, linear canonical, and Haar wavelet transforms

Abstract: Fractional Fourier Transform are introduced as sparsifying transforms.Linear Canonical Transforms are introduced as sparsifying transforms.Various approaches for compressing three-dimensional images are suggested. Display Omitted Sparse recovery aims to reconstruct signals that are sparse in a linear transform domain from a heavily underdetermined set of measurements. The success of sparse recovery relies critically on the knowledge of transform domains that give compressible representations of the signal of interest. Here we consider two- and three-dimensional images, and investigate various multi-dimensional transforms in terms of the compressibility of the resultant coefficients. Specifically, we compare the fractional Fourier (FRT) and linear canonical transforms (LCT), which are generalized versions of the Fourier transform (FT), as well as Hartley and simplified fractional Hartley transforms, which differ from corresponding Fourier transforms in that they produce real outputs for real inputs. We also examine a cascade approach to improve transform-domain sparsity, where the Haar wavelet transform is applied following an initial Hartley transform. To compare the various methods, images are recovered from a subset of coefficients in the respective transform domains. The number of coefficients that are retained in the subset are varied systematically to examine the level of signal sparsity in each transform domain. Recovery performance is assessed via the structural similarity index (SSIM) and mean squared error (MSE) in reference to original images. Our analyses show that FRT and LCT transform yield the most sparse representations among the tested transforms as dictated by the improved quality of the recovered images. Furthermore, the cascade approach improves transform-domain sparsity among techniques applied on small image patches.

Discrete Linear Canonical Transform Based on Hyperdifferential Operators

Abstract: Linear canonical transforms (LCTs) are of importance in many areas of science and engineering with many applications. Therefore, a satisfactory discrete implementation is of considerable interest. Although there are methods that link the samples of the input signal to the samples of the linear canonical transformed output signal, no widely-accepted definition of the discrete LCT has been established. We introduce a new approach to defining the discrete linear canonical transform (DLCT) by employing operator theory. Operators are abstract entities that can have both continuous and discrete concrete manifestations. Generating the continuous and discrete manifestations of LCTs from the same abstract operator framework allows us to define the continuous and discrete transforms in a structurally analogous manner. By utilizing hyperdifferential operators, we obtain a DLCT matrix, which is totally compatible with the theory of the discrete Fourier transform (DFT) and its dual and circulant structure, which makes further analytical manipulations and progress possible. The proposed DLCT is to the continuous LCT, what the DFT is to the continuous Fourier transform. The DLCT of the signal is obtained simply by multiplying the vector holding the samples of the input signal by the DLCT matrix.

Straggler Resilient Serverless Computing Based on Polar Codes

Abstract: We propose a serverless computing mechanism for distributed computation based on polar codes. Serverless computing is an emerging cloud based computation model that lets users run their functions on the cloud without provisioning or managing servers. Our proposed approach is a hybrid computing framework that carries out computationally expensive tasks such as linear algebraic operations involving large-scale data using serverless computing and does the rest of the processing locally. We address the limitations and reliability issues of serverless platforms such as straggling workers using coding theory, drawing ideas from recent literature on coded computation. The proposed mechanism uses polar codes to ensure straggler-resilience in a computationally effective manner. We provide extensive evidence showing polar codes outperform other coding methods. We have designed a sequential decoder specifically for polar codes in erasure channels with full-precision input and outputs. In addition, we have extended the proposed method to the matrix multiplication case where both matrices being multiplied are coded. The proposed coded computation scheme is implemented for AWS Lambda. Experiment results are presented where the performance of the proposed coded computation technique is tested in optimization via gradient descent. Finally, we introduce the idea of partial polarization which reduces the computational burden of encoding and decoding at the expense of straggler-resilience.

Erasure coding for distributed storage: an overview

Abstract: In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.

Repairing Reed-Solomon Codes With Multiple Erasures

Abstract: Despite their exceptional error-correcting properties, Reed–Solomon (RS) codes have been overlooked in distributed storage applications due to the common belief that they have poor repair bandwidth. A naive repair approach would require for the whole file to be reconstructed in order to recover a single erased codeword symbol. In a recent work, Guruswami and Wootters (STOC’16) proposed a single erasure repair method for RS codes that achieves the optimal repair bandwidth amongst all linear encoding schemes. Their key idea is to recover the erased symbol by collecting a sufficiently large number of its traces, each of which can be constructed from a number of traces of other symbols. We extend the trace collection technique to cope with two and three erasures.

Exact expressions for double descent and implicit regularization via surrogate random design

Abstract: Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We provide the first exact non-asymptotic expressions for double descent of the minimum norm linear estimator. Our approach involves constructing a special determinantal point process which we call surrogate random design, to replace the standard i.i.d. design of the training sample. This surrogate design admits exact expressions for the mean squared error of the estimator while preserving the key properties of the standard design. We also establish an exact implicit regularization result for over-parameterized training samples. In particular, we show that, for the surrogate design, the implicit bias of the unregularized minimum norm estimator precisely corresponds to solving a ridge-regularized least squares problem on the population distribution. In our analysis we introduce a new mathematical tool of independent interest: the class of random matrices for which determinant commutes with expectation.

Revealing the Structure of Deep Neural Networks via Convex Duality

Abstract: We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove that norm regularized deep ReLU networks yield spline interpolation for one-dimensional datasets which was previously known only for two-layer networks. Furthermore, we provide closed-form solutions for the optimal layer weights when data is rank-one or whitened. The same analysis also applies to architectures with batch normalization even for arbitrary data. Therefore, we obtain a complete explanation for a recent empirical observation termed Neural Collapse where class means collapse to the vertices of a simplex equiangular tight frame.

Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain

Abstract: The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. Many theories for this transform are already known, but the uniform sampling theorem, as well as the sampling rate conversion theory about arbitrary lattices sampling in the LCT domain are still to be determined. Focusing on these issues, this paper carefully investigates arbitrary lattices sampling, the sampling with separable matrices and nonseparable matrices, to obtain uniform sampling theorem and the sampling rate conversion theory in the LCT domain. Firstly, the spectral expression of the discrete-time signal sampled via arbitrary lattice is deduced in the LCT domain. Based on it we propose the alias-free sampling relationship between two matrices and present the perfect reconstruction expressions for bandlimited signals in the LCT domain. Secondly, for further research on discrete signals to obtain sampling rate conversion theory, we define the multidimensional discrete time linear canonical transform (MDTLCT), as well as the convolution for the MDTLCT. Thirdly, the formulas of multidimensional interpolation and decimation via integer matrices in the LCT domain are derived. Then, based on the results of interpolation and decimation, we make analyses of the sampling rate conversion via rational matrices in the LCT domain, including spectral analyses and the formulas in time domain. Finally, simulation results and the potential applications of the theories are also presented.