Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length $n$ and distance $d = O(1)$, one can correct $\operatorname{polylog}(n)$ random errors in $\operatorname{poly}(n)$ time (which is well beyond the worst-case error tolerance of $O(1)$). In this paper, we consider the problem of `syndrome decoding' Reed-Muller codes from random errors. More specifically, given the $\operatorname{polylog}(n)$-bit long syndrome vector of a codeword corrupted in $\operatorname{polylog}(n)$ random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in $\operatorname{polylog}(n)$ time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.Comment: 24 page

A Spectral Bound on Hypergraph Discrepancy

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the discrepancy of $\mathcal{H}$ is $O(\sqrt{t} + \lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m \geq n$ edges is almost surely $O(\sqrt{t})$ as $n$ grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1811.01491, several changes to the presentatio

Renal and Cardiovascular Outcomes Associated with Cannabis Use in Veterans with Advanced Chronic Kidney Disease Transitioning to Dialysis

Background: Legalization of cannabis and its constituents may lead to increased exposure of a higher number of consumers to cannabis use, including those with chronic kidney disease (CKD). We expect increase in cannabis use, especially, in patients with compromised kidney function. However, there is sparse literature on the effects of cannabis use on cardiorenal and mortality outcomes among patients with advanced CKD. Objectives: The main goal of the current study was to examine the effect of cannabis use on renal and cerebrovascular outcomes as well as mortality in patients with advanced CKD. Using a nationwide cohort of US veterans with advanced CKD transitioning to dialysis, we were guided by the following aims: Aim 1) Evaluate the effect of cannabis exposure on kidney function by 1a) examining the association of cannabis exposure with progression of CKD, and 1b) investigating the association of cannabis exposure with the incidence of acute kidney injury (AKI); Aim 2) Investigate the association of cannabis exposure with the incidence of stroke; and Aim 3) Examine the association of cannabis exposure with mortality (mortality due to any reason and cardiovascular mortality). Methods: We used a retrospective cohort study design in a nationally representative cohort of US veterans with incident End-Stage Renal Disease (ESRD) who transitioned to renal replacement therapy from October 1, 2007 through March 31, 2015. The Transition of Care in Chronic Kidney Disease (TC-CKD) cohort consisted of 102,477 US veterans with incident ESRD identified from the US Renal Data System (USRDS). Urine toxicology tests (UTOX) determined the use of cannabis, opioids, other drugs, and combinations of the same in the patients who had undergone a UTOX test within the year prior to dialysis initiation. After applying inclusion and exclusion criteria, 7,146 patients comprised our study population. Chapter 2 discusses our examination of the association between UTOX groups and renal outcomes, including both long-term changes in estimated glomerular filtration rate (eGFR) and the incidence of AKI. We used mixed-effects models with random intercepts and slopes and logistic regression to examine the association between UTOX groups and the risk of change in eGFR and AKI, respectively. Chapter 3 describes our investigation of the association between UTOX groups and cerebrovascular accident (CVA) events using Cox proportional hazard models. Finally, Chapter 4 presents our research on the association between UTOX groups and mortality using Cox proportional hazard models (mortality due to any reason) and Fine and Grey’s competing risk regression (cardiovascular mortality). Results: Cannabis users were more likely to be younger (57 years cannabis users vs. 60 years no drug use), less likely to be white (45% cannabis users vs. 55% no drug use), and more likely to be smokers (69% cannabis users vs. 38% no drug use). We observed that cannabis use alone or combined with opioids or other drugs (vs. no drug use) was not significantly associated with steeper eGFR slopes or risk of AKI (P-value 0.4-1.0). We also found that the use of cannabis alone, opioids, other drugs alone, or a combination of these (vs. no drug use) was not significantly associated with the risk of CVA events (P-value 0.4-1.0). Finally, cannabis use alone or combined use of cannabis with opioids or with other drugs (vs. no drug use) was not significantly associated with mortality due to any reason or CV mortality (P-value 0.4-1.0). Conclusion: This study is the first, to our knowledge, to ascertain use of cannabis, opioids, or other drugs via UTOX tests to examine the association between various combinations of exposures and renal/cardiovascular/mortality outcomes in patients with advanced CKD transitioning to dialysis. The study findings suggest the absence of a harmful association between exposure to cannabis and renal/cerebrovascular/mortality outcomes. Future clinical trials and further epidemiological studies are needed to confirm these findings and further expand our understanding of the health effects of cannabis use in the general population as well as in patients with compromised kidney function

Pre-ESRD Depression and Post-ESRD Mortality in Patients with Advanced CKD Transitioning to Dialysis.

Background and objectivesDepression in patients with nondialysis-dependent CKD is often undiagnosed, empirically overlooked, and associated with higher risk of death, progression to ESRD, and hospitalization. However, there is a paucity of evidence on the association between the presence of depression in patients with advanced nondialysis-dependent CKD and post-ESRD mortality, particularly among those in the transition period from late-stage nondialysis-dependent CKD to maintenance dialysis.Design, setting, participants, & measurementsFrom a nation-wide cohort of 45,076 United States veterans who transitioned to ESRD over 4 contemporary years (November of 2007 to September of 2011), we identified 10,454 (23%) patients with a depression diagnosis during the predialysis period. We examined the association of pre-ESRD depression with all-cause mortality after transition to dialysis using Cox proportional hazards models adjusted for sociodemographics, comorbidities, and medications.ResultsPatients were 72±11 years old (mean±SD) and included 95% men, 66% patients with diabetes, and 23% blacks. The crude mortality rate was similar in patients with depression (289/1000 patient-years; 95% confidence interval, 282 to 297) versus patients without depression (286/1000 patient-years; 95% confidence interval, 282 to 290). Compared with patients without depression, patients with depression had a 6% higher all-cause mortality risk in the adjusted model (hazard ratio, 1.06; 95% confidence interval, 1.03 to 1.09). Similar results were found across all selected subgroups as well as in sensitivity analyses using alternate definitions of depression.ConclusionPre-ESRD depression has a weak association with post-ESRD mortality in veterans transitioning to dialysis