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Magnetic recording devices are still competitive in the storage density race with solid-state devices thanks to new technologies such as two-dimensional magnetic recording (TDMR). Advanced data processing schemes are needed to guarantee reliability in TDMR. Data patterns where a bit is surrounded by complementary bits at the four positions with Manhattan distance $1$ on the TDMR grid are called plus isolation (PIS) patterns, and they are error-prone. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely optimal plus LOCO (OP-LOCO) codes, that prevent these patterns from being written in a TDMR device. However, in the high-density regime or the low-energy regime, additional error-prone patterns emerge, specifically data patterns where a bit is surrounded by complementary bits at only three positions with Manhattan distance $1$, and we call them incomplete plus isolation (IPIS) patterns. In this paper, we present capacity-achieving codes that forbid both PIS and IPIS patterns in TDMR systems with wide read heads. We collectively call the PIS and IPIS patterns rotated T isolation (RTIS) patterns, and we call the new codes optimal T LOCO (OT-LOCO) codes. We analyze OT-LOCO codes and present their simple encoding-decoding rule that allows reconfigurability. We also present a novel bridging idea for these codes to further increase the rate. Our simulation results demonstrate that OT-LOCO codes are capable of eliminating media noise effects entirely at practical TD densities with high rates. To further preserve the storage capacity, we suggest using OP-LOCO codes early in the device lifetime, then employing the reconfiguration property to switch to OT-LOCO codes later. While the point of reconfiguration on the density/energy axis is decided manually at the moment, the next step is to use machine learning to take that decision based on the TDMR device status., Comment: 15 pages (double column), 11 figures, submitted to the IEEE Transactions on Magnetics (TMAG). arXiv admin note: text overlap with arXiv:2010.10686

Low-density parity-check (LDPC) codes are specified by graphs, and are the error correction technique of choice in many communications and data storage contexts. Message passing decoders diffuse information carried by parity bits into the payload, and this paper measures the value of engineering parity bits to be more reliable than message bits. We consider the binary symmetric channel (BSC) and measure the impact of unequal data protection on the threshold of a regular LDPC code. Our analysis also includes doping where the parity bits are known to the decoder. We investigate BSC with Gallager-A decoder, with a $3$-level-alphabet decoder, and with a full belief propagation decoder. We demonstrate through theoretical analysis and simulation that non-equiprobable inputs lead to significant improvements both in the threshold and in the speed with which the decoder converges. We also show that all these improvements are possible even with a simple $3$-level-alphabet decoder., 8 pages (double column), 1 figure, submitted to the International Symposium on Topics in Coding (ISTC)

The pivotal storage density win achieved by solid-state devices over magnetic devices recently is a result of multiple innovations in physics, architecture, and signal processing. Constrained coding is used in Flash devices to increase reliability via mitigating inter-cell interference. Recently, capacity-achieving constrained codes were introduced to serve that purpose. While these codes result in minimal redundancy, they result in non-negligible complexity increase and access speed limitation since pages cannot be read separately. In this paper, we suggest new constrained coding schemes that have low-complexity and preserve the desirable high access speed in modern Flash devices. The idea is to eliminate error-prone patterns by coding data either only on the left-most page (binary coding) or only on the two left-most pages ($4$-ary coding) while leaving data on all the remaining pages uncoded. Our coding schemes are systematic and capacity-approaching. We refer to the proposed schemes as read-and-run (RR) constrained coding schemes. The $4$-ary RR coding scheme is introduced to limit the rate loss. We analyze the new RR coding schemes and discuss their impact on the probability of occurrence of different charge levels. We also demonstrate the performance improvement achieved via RR coding on a practical triple-level cell Flash device., 30 pages (single column), 5 figures, submitted to the IEEE Transactions on Communications (TCOM). arXiv admin note: substantial text overlap with arXiv:2111.07415

Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code $\mathcal{C}$, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator $U_Z$ to preserve $\mathcal{C}$ and induce $U_L$. The group of $Z$-stabilizers in a CSS code $\mathcal{C}$ is determined by the dual of a classical $[n, k_1]$ binary code $\mathcal{C}_1$, and the group of $X$-stabilizers is determined by a classical $[n, k_2]$ binary code $\mathcal{C}_2$ that is contained in $\mathcal{C}_1$. The requirement that a diagonal physical operator $U_Z$ fixes a CSS code $\mathcal{C}$ leads to constraints on the congruence of weights in cosets of $\mathcal{C}_2$. These constraints are a perfect fit to divisible codes, and represent an opportunity to take advantage of the extensive literature on classical codes with two or three weights. We construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms. We provide a simple alternative to the standard method of deriving the coset weight distributions (based on Dickson normal form) that may be of independent interest. Finally, we develop an approach to circumventing the Eastin-Knill Theorem which states that no QECC can implement a universal set of logical gates through transversal gates alone. The essential idea is to design stabilizer codes in layers, with $N_1$ inner qubits and $N_2$ outer qubits, and to assemble a universal set of fault tolerant gates on the inner qubits., Jingzhen Hu and Qingzhong Liang contribute equally to this work. 11 pages. Comments welcome!

Phase retrieval is known to always be unstable when using a frame or continuous frame for an infinite dimensional Hilbert space. We consider a generalization of phase retrieval to the setting of subspaces of $L_2$ which coincides with using a continuous frame for phase retrieval when the subspace is the range of the analysis operator of a continuous frame. We then prove that there do exist infinite dimensional subspaces of $L_2$ where phase retrieval is stable. That is, we give a method for constructing an infinite dimensional subspace $Y\subseteq L_2$ such that there exists $C\geq 1$ so that $$\min\big(\big\|f-g\big\|_{L_2},\big\|f+g\big\|_{L_2}\big)\leq C \big\| |f|-|g| \big\|_{L_2} \qquad\textrm{ for all }f,g\in Y. $$ This construction also leads to new results on uniform stability of phase retrieval in finite dimensions. Our construction has a deterministic component and a random component. When using sub-Gaussian random variables we achieve phase retrieval with high probability and stability constant independent of the dimension $n$ when using $m$ on the order of $n$ random vectors. Without sub-Gaussian or any other higher moment assumptions, we are able to achieve phase retrieval with high probability and stability constant independent of the dimension $n$ when using $m$ on the order of $n\log(n)$ random vectors., 27 pages

Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of $Z$-stabilizers, and addition of $X$-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level $l$ inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level $l+1$ induces the original logical gate. The next step is judicious removal of $Z$-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level $l$ to level $l+1$, each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of $Z$-stabilizers may reduce distance, and the purpose of the third basic operation, addition of $X$-stabilizers, is to compensate for such losses. For the coherent noise model, we describe how to switch between computation and storage of intermediate results in a decoherence-free subspace by simply applying Pauli $X$ matrices. The approach to logical gate synthesis taken in prior work focuses on the code states, and results in sufficient conditions for a CSS code to be fixed by a transversal $Z$-rotation. In contrast, we derive necessary and sufficient conditions by analyzing the action of a transversal diagonal gate on the stabilizer group that determines the code. The power of our approach is demonstrated by two proofs of concept: the $[[2^{l+1}-2,2,2]]$ triorthogonal code family, and the $[[2^m,\binom{m}{r},2^{\min\{r,m-r\}}]]$ quantum Reed-Muller code family., Jingzhen Hu and Qingzhong Liang contribute equally to this work. 15 pages, two columns, 8 figures, and 1 table. Comments welcome! This work is an application based on the mathematical framework introduced in arXiv:2109.13481

Spatially-coupled (SC) codes, known for their threshold saturation phenomenon and low-latency windowed decoding algorithms, are ideal for streaming applications and data storage systems. SC codes are constructed by partitioning an underlying block code, followed by rearranging and concatenating the partitioned components in a convolutional manner. The number of partitioned components determines the memory of SC codes. In this paper, we investigate the relation between the performance of SC codes and the density distribution of partitioning matrices. While adopting higher memories results in improved SC code performance, obtaining finite-length, high-performance SC codes with high memory is known to be computationally challenging. We break this computational bottleneck by developing a novel probabilistic framework that obtains (locally) optimal density distributions via gradient descent. Starting from random partitioning matrices abiding by the obtained distribution, we perform low-complexity optimization algorithms that minimize the number of detrimental objects to construct high-memory, high-performance quasi-cyclic SC codes. We apply our framework to various objects of interests, from the simplest short cycles, to more sophisticated objects such as concatenated cycles aiming at finer-grained optimization. Simulation results show that codes obtained through our proposed method notably outperform state-of-the-art SC codes with the same constraint length and optimized SC codes with uniform partitioning. The performance gain is shown to be universal over a variety of channels, from canonical channels such as additive white Gaussian noise and binary symmetric channels, to practical channels underlying flash memory and magnetic recording systems., 26 pages, 30 figures, 2 tables, the manuscript has been submitted to IEEE Transactions on Information Theory. arXiv admin note: substantial text overlap with arXiv:2101.09962

Computational storage, known as a solution to significantly reduce the latency by moving data-processing down to the data storage, has received wide attention because of its potential to accelerate data-driven devices at the edge. To meet the insatiable appetite for complicated functionalities tailored for intelligent devices such as autonomous vehicles, properties including heterogeneity, scalability, and flexibility are becoming increasingly important. Based on our prior work on hierarchical erasure coding that enables scalability and flexibility in cloud storage, we develop an efficient decoding algorithm that corrects a mixture of errors and erasures simultaneously. We first extract the basic component code, the so-called extended Cauchy (EC) codes, of the proposed coding solution. We prove that the class of EC codes is strictly larger than that of relevant codes with known explicit decoding algorithms. Motivated by this finding, we then develop an efficient decoding method for the general class of EC codes, based on which we propose the local and global decoding algorithms for the hierarchical codes. Our proposed hybrid error correction not only enables the usage of hierarchical codes in computational storage at the edge, but also applies to any Cauchy-like codes and allows potentially wider applications of the EC codes., 29 pages (single column), 0 figures, to be submitted to IEEE Transactions on Communications

Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across the space and the scaling group, which is shown to be both sufficient and necessary to achieve equivariance for the regular representation of the scaling-translation group ST . To reduce the model complexity and computational burden, we decompose the convolutional filters under two pre-fixed separable bases and truncate the expansion to low-frequency components. A further benefit of the truncated filter expansion is the improved deformation robustness of the equivariant representation, a property which is theoretically analyzed and empirically verified. Numerical experiments demonstrate that the proposed scaling-translation-equivariant network with decomposed convolutional filters (ScDCFNet) achieves significantly improved performance in multiscale image classification and better interpretability than regular CNNs at a reduced model size.

Deep-learning-based models have been successfully applied to the problem of detecting fake news on social media. While the correlations among news articles have been shown to be effective cues for online news analysis, existing deep-learning-based methods often ignore this information and only consider each news article individually. To overcome this limitation, we develop a graph-theoretic method that inherits the power of deep learning while at the same time utilizing the correlations among the articles. We formulate fake news detection as an inference problem in a Markov random field (MRF) which can be solved by the iterative mean-field algorithm. We then unfold the mean-field algorithm into hidden layers that are composed of common neural network operations. By integrating these hidden layers on top of a deep network, which produces the MRF potentials, we obtain our deep MRF model for fake news detection. Experimental results on well-known datasets show that the proposed model improves upon various state-of-the-art models.

To assess changes in quality of care for children at risk for autism spectrum disorders (ASD) due to process improvement and implementation of a digital screening form.The process of screening for ASD was studied in an academic primary care pediatrics clinic before and after implementation of a digital version of the Modified Checklist for Autism in Toddlers - Revised with Follow-up with automated risk assessment. Quality metrics included accuracy of documentation of screening results and appropriate action for positive screens (secondary screening or referral). Participating physicians completed pre- and postintervention surveys to measure changes in attitudes toward feasibility and value of screening for ASD. Evidence of change was evaluated with statistical process control charts and χAccurate documentation in the electronic health record of screening results increased from 54% to 92% (38% increase, 95% CI 14%-64%) and appropriate action for children screening positive increased from 25% to 85% (60% increase, 95% CI 35%-85%). A total of 90% of participating physicians agreed that the transition to a digital screening form improved their clinical assessment of autism risk.Implementation of a tablet-based digital version of the Modified Checklist for Autism in Toddlers - Revised with Follow-up led to improved quality of care for children at risk for ASD and increased acceptability of screening for ASD. Continued efforts towards improving the process of screening for ASD could facilitate rapid, early diagnosis of ASD and advance the accuracy of studies of the impact of screening.

Racetrack memory is a non-volatile memory engineered to provide both high density and low latency, that is subject to synchronization or shift errors. This paper describes a fast coding solution, in which delimiter bits assist in identifying the type of shift error, and easily implementable graph-based codes are used to correct the error, once identified. A code that is able to detect and correct double shift errors is described in detail., Technical report 2017

In this chapter, we present a simple classification scheme that utilizes only 1-bit measurements of the training and testing data. Our method is intended to be efficient in terms of computation and storage while also allowing for a rigorous mathematical analysis. After providing some motivation, we present our method and analyze its performance for a simple data model. We also discuss extensions of the method to the hierarchical data setting, and include some further implementation considerations. Experimental evidence provided in this chapter demonstrates that our methods yield accurate classification on a variety of synthetic and real data.

Deep neural networks have proved very successful in domains where large training sets are available, but when the number of training samples is small, their performance suffers from overfitting. Prior methods of reducing overfitting such as weight decay, Dropout and DropConnect are data-independent. This paper proposes a new method, GraphConnect, that is data-dependent, and is motivated by the observation that data of interest lie close to a manifold. The new method encourages the relationships between the learned decisions to resemble a graph representing the manifold structure. Essentially GraphConnect is designed to learn attributes that are present in data samples in contrast to weight decay, Dropout and DropConnect which are simply designed to make it more difficult to fit to random error or noise. Empirical Rademacher complexity is used to connect the generalization error of the neural network to spectral properties of the graph learned from the input data. This framework is used to show that GraphConnect is superior to weight decay. Experimental results on several benchmark datasets validate the theoretical analysis, and show that when the number of training samples is small, GraphConnect is able to significantly improve performance over weight decay., Theorems need more validation

Guesswork forms the mathematical framework for quantifying computational security subject to brute-force determination by query. In this paper, we consider guesswork subject to a per-symbol Shannon entropy budget. We introduce inscrutability rate to quantify the asymptotic difficulty of guessing U out of V secret strings drawn from the string-source and prove that the inscrutability rate of any string-source supported on a finite alphabet X, if it exists, lies between the per-symbol Shannon entropy constraint and log |X|. We show that for a stationary string-source, the inscrutability rate of guessing any fraction (1 - ϵ) of the V strings for any fixed ϵ > 0, as V grows, approaches the per-symbol Shannon entropy constraint (which is equal to the Shannon entropy rate for the stationary string-source). This corresponds to the minimum inscrutability rate among all string-sources with the same per-symbol Shannon entropy. We further prove that the inscrutability rate of any finite-order Markov string-source with hidden statistics remains the same as the unhidden case, i.e., the asymptotic value of hiding the statistics per each symbol is vanishing. On the other hand, we show that there exists a string-source that achieves the upper limit on the inscrutability rate, i.e., log |X|, under the same Shannon entropy budget.

We present fundamental limits on the reliable classification of linear and affine subspaces from noisy, linear features. Drawing an analogy between discrimination among subspaces and communication over vector wireless channels, we propose two Shannon-inspired measures to characterize asymptotic classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient conditions for the misclassification probability to vanish as the signal dimension, the number of features, and the number of subspaces to be discerned all approach infinity. Second, we define the diversity-discrimination tradeoff which, by analogy with the diversity-multiplexing tradeoff of fading vector channels, characterizes relationships between the number of discernible subspaces and the misclassification probability as the noise power approaches zero. We derive upper and lower bounds on these measures which are tight in many regimes. Numerical results, including a face recognition application, validate the results in practice., 19 pages, 4 figures. Revised submission to IEEE Transactions on Information Theory

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods -- the spatial complement and the Naimark complement -- for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given $M, N, m \in \NN$ and $\{\lambda_j\}_{j=1}^M$, does there exist a fusion frame in $\RR^M$ with $N$ subspaces of dimension $m$ for which $\{\lambda_j\}_{j=1}^M$ are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters $M, N, m \in \NN$ and $\{\lambda_j\}_{j=1}^M$. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval., Comment: 24 pages

Compressed Sensing aims to capture attributes of a sparse signal using very few measurements. Cand\`{e}s and Tao showed that sparse reconstruction is possible if the sensing matrix acts as a near isometry on all $\boldsymbol{k}$-sparse signals. This property holds with overwhelming probability if the entries of the matrix are generated by an iid Gaussian or Bernoulli process. There has been significant recent interest in an alternative signal processing framework; exploiting deterministic sensing matrices that with overwhelming probability act as a near isometry on $\boldsymbol{k}$-sparse vectors with uniformly random support, a geometric condition that is called the Statistical Restricted Isometry Property or StRIP. This paper considers a family of deterministic sensing matrices satisfying the StRIP that are based on \srm codes (binary chirps) and a $\boldsymbol{k}$-sparse reconstruction algorithm with sublinear complexity. In the presence of stochastic noise in the data domain, this paper derives bounds on the $\boldsymbol{\ell_2}$ accuracy of approximation in terms of the $\boldsymbol{\ell_2}$ norm of the measurement noise and the accuracy of the best $\boldsymbol{k}$-sparse approximation, also measured in the $\boldsymbol{\ell_2}$ norm. This type of $\boldsymbol{\ell_2 /\ell_2}$ bound is tighter than the standard $\boldsymbol{\ell_2 /\ell_1}$ or $\boldsymbol{\ell_1/ \ell_1}$ bounds.