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

Testing Whether MRP4 (a cAMP Efflux Pump) and the Beta 2 Andrenergic Receptor (an Upstream Regulator of cAMP Signaling Pathways

Background and Aim: MRP4/ABCC4 is an ABC transporter that can efflux the second-messenger, cAMP, from cells. MRP4 has a PDZ interacting motif at its carboxy terminal end through which it binds to scaffolding proteins NHERF1 and PDZK1. Previous studies have shown that PDZK1 serves as a scaffold physically coupling MRP4 with the cystic fibrosis transmembrane conductance regulator (CFTR). This protein complex functionally couples cAMP regulation of CFTR function with MRP4 cAMP transporter activity [Li, C., et al., Spatiotemporal coupling of cAMP transporter to CFTR chloride channel function in the gut epithelia. Cell, 2007. 131(5): p. 940-51]. We hypothesized that the MRP4 PDZ domain can bind MRP4 to scaffolding proteins other than NHERF1 and PDZK1 and that those PDZ proteins serve to physically and functionally link MRP4 to other proteins involved in cAMP signaling. High expression of MRP4 has been observed in normal prostate and in human prostate cancer cell lines such as LNCaP cells [Cai, C., et al., Androgen induces expression of the multidrug resistance protein gene MRP4 in prostate cancer cells. Prostate Cancer Prostatic Dis, 2007. 10(1): p. 39-45]. In these cells, there are reports that the !2 adrenergic receptor (!2AR), a G protein-coupled receptor that ultimately signals through cAMP, is also highly expressed [Kasbohm, E.A., et al., Androgen receptor activation by G(s) signaling in prostate cancer cells. J Biol Chem, 2005. 280(12): p. 11583-9]. !2AR also has a PDZ interacting motif at its carboxy terminal end. We hypothesized that in these cells there is the possibility of interaction between MRP4 and !2AR through a shared PDZ protein, leading to physical and functional association of these proteins. Methodology: We probed Panomics PDZ protein arrays with biotinylated MRP4 peptides consisting of MRP4 PDZ interacting motif, and identified potential PDZ partner proteins to which MRP4 peptide binds. We used LNCaP cells to quantify the mRNA expression of these putative PDZ partner proteins, as well as MRP4 and !2AR. We carried out pull down assays to test for physical association between the three proteins. We used a cAMP reporter assay to test whether activation of !2AR induced cAMP signaling and determine whether this signaling was modulated by MRP4 expression. Results: Out of 93 PDZ domains, MRP4 showed interaction with 24 PDZ domains. The five candidate PDZ proteins chosen for further studies met the following criteria: (a) They were documented in the literature to interact functionally with GPCRs (G protein-coupled receptors) that upon stimulation couple with G proteins and cause a rise in intracellular cAMP, and (b) They were expressed in prostate cells and co-localized with MRP4. Based on previous studies, only two PDZ domains, NHERF1 and MAGI3, were reported to bind with G s, which is a subunit responsible for production of cAMP in response to activation of certain type of GPCRs. LNCaP cells have higher expression of MRP4 and !2AR (a G protein-coupled receptor that binds G s subunit). Using pull down assays, we showed a physical association between MRP4 and !2AR. We also showed a functional association between !2AR and MRP4, because inhibition of MRP4 modulated !2AR-induced cAMP signaling in LNCaP cells. Conclusion: MRP4 is physically and functionally associated with the !2AR in LNCaP cells. This association may be facilitated by a scaffolding protein, which may be MAGI3. This protein may be responsible for holding the macromolecular complex of two proteins, MRP4 and !2AR. This association may be important for regulating cAMP levels in LNCaP cells and affecting the downstream expression of certain genes that depend on the cAMP signal transduction