Estimating Renyi Entropy of Discrete Distributions

It was recently shown that estimating the Shannon entropy $H({\rm p})$ of a discrete $k$-symbol distribution ${\rm p}$ requires $\Theta(k/\log k)$ samples, a number that grows near-linearly in the support size. In many applications $H({\rm p})$ can be replaced by the more general R\'enyi entropy of order $\alpha$, $H_\alpha({\rm p})$. We determine the number of samples needed to estimate $H_\alpha({\rm p})$ for all $\alpha$, showing that $\alpha < 1$ requires a super-linear, roughly $k^{1/\alpha}$ samples, noninteger $\alpha>1$ requires a near-linear $k$ samples, but, perhaps surprisingly, integer $\alpha>1$ requires only $\Theta(k^{1-1/\alpha})$ samples. Furthermore, developing on a recently established connection between polynomial approximation and estimation of additive functions of the form $\sum_{x} f({\rm p}_x)$, we reduce the sample complexity for noninteger values of $\alpha$ by a factor of $\log k$ compared to the empirical estimator. The estimators achieving these bounds are simple and run in time linear in the number of samples. Our lower bounds provide explicit constructions of distributions with different R\'enyi entropies that are hard to distinguish

Homocysteine and Hypertension in Diabetes: Does PPARγ Have a Regulatory Role?

Dysfunction of macro- and microvessels is a major cause of morbidity and mortality in patients with cardio-renovascular diseases such as atherosclerosis, hypertension, and diabetes. Renal failure and impairment of renal function due to vasoconstriction of the glomerular arteriole in diabetic nephropathy leads to renal volume retention and increase in plasma homocysteine level. Homocysteine, which is a nonprotein amino acid, at elevated levels is an independent cardio-renovascular risk factor. Homocysteine induces oxidative injury of vascular endothelial cells, involved in matrix remodeling through modulation of the matrix metalloproteinase (MMP)/tissue inhibitor of metalloproteinase (TIMP) axis, and increased formation and accumulation of extracellular matrix protein, such as collagen. In heart this leads to increased endothelial-myocyte uncoupling resulting in diastolic dysfunction and hypertension. In the kidney, increased matrix accumulation in the glomerulus causes glomerulosclerosis resulting in hypofiltration, increased renal volume retention, and hypertension. PPARγ agonist reduces tissue homocysteine levels and is reported to ameliorate homocysteine-induced deleterious vascular effects in diabetes. This review, in light of current information, focuses on the beneficial effects of PPARγ agonist in homocysteine-associated hypertension and vascular remodeling in diabetes

Homocysteine and reactive oxygen species in metabolic syndrome, type 2 diabetes mellitus, and atheroscleropathy: The pleiotropic effects of folate supplementation

Homocysteine has emerged as a novel independent marker of risk for the development of cardiovascular disease over the past three decades. Additionally, there is a graded mortality risk associated with an elevated fasting plasma total homocysteine (tHcy). Metabolic syndrome (MS) and type 2 diabetes mellitus (T2DM) are now considered to be a strong coronary heart disease (CHD) risk enhancer and a CHD risk equivalent respectively. Hyperhomocysteinemia (HHcy) in patients with MS and T2DM would be expected to share a similar prevalence to the general population of five to seven percent and of even greater importance is: Declining glomerular filtration and overt diabetic nephropathy is a major determinant of tHcy elevation in MS and T2DM. There are multiple metabolic toxicities resulting in an excess of reactive oxygen species associated with MS, T2DM, and the accelerated atherosclerosis (atheroscleropathy). HHcy is associated with an increased risk of cardiovascular disease, and its individual role and how it interacts with the other multiple toxicities are presented. The water-soluble B vitamins (especially folate and cobalamin-vitamin B(12)) have been shown to lower HHcy. The absence of the cystathionine beta synthase enzyme in human vascular cells contributes to the importance of a dual role of folic acid in lowering tHcy through remethylation, as well as, its action of being an electron and hydrogen donor to the essential cofactor tetrahydrobiopterin. This folate shuttle facilitates the important recoupling of the uncoupled endothelial nitric oxide synthase enzyme reaction and may restore the synthesis of the omnipotent endothelial nitric oxide to the vasculature

Intimal redox stress: Accelerated atherosclerosis in metabolic syndrome and type 2 diabetes mellitus. Atheroscleropathy

Metabolic syndrome, insulin resistance, prediabetes, and overt type 2 diabetes mellitus are associated with an accelerated atherosclerosis (atheroscleropathy). This quartet is also associated with multiple metabolic toxicities resulting in the production of reactive oxygen species. The redox stress associated with these reactive oxygen species contribute to the development, progression, and the final fate of the arterial vessel wall in prediabetic and diabetic atheroscleropathy. The prevention of morbidity and mortality of these intersecting metabolic diseases can be approached through comprehensive global risk reduction

Isolated low high density lipoprotein-cholesterol (HDL-C): implications of global risk reduction. Case report and systematic scientific review

BACKGROUND: The importance of low high-density lipoprotein cholesterol (HDL-C), elevated non HDL-C (as part of the metabolic syndrome, prediabetes, and type 2 diabetes mellitus), and an isolated low HDL-C is rapidly emerging. The antiatherosclerotic roles of reverse cholesterol transport and the pleiotropic antioxidant – anti-inflammatory mechanistic effects of HDL-C are undergoing rapid exponential growth. CASE PRESENTATION: In 1997 a 53-year-old Caucasian male presented with a lipoprotein profile of many years duration with an isolated low HDL-C and uric acid levels in the upper quintile of normal. He developed an acute myocardial infarction involving the right coronary artery and had percutaneous transluminal coronary angioplasty with stenting of this lesion. He also demonstrated a non-critical non-flow limiting lesion of the proximal left anterior descending coronary artery at the time of this evaluation. Following a program of global risk reduction this patient has done well over the past 7 years and remains free of any clinical signs and symptoms of atherosclerosis. His HDL-C and uric acid levels are currently in the normal physiological range. CONCLUSION: Low HDL-C and isolated low HDL-C constitute an important risk factor for atherosclerosis. Therapies that lead to a return to normal physiologic range of HDL-C may result in the delay of atherosclerotic progression

Uric acid: A new look at an old risk marker for cardiovascular disease, metabolic syndrome, and type 2 diabetes mellitus: The urate redox shuttle

BACKGROUND: The topical role of uric acid and its relation to cardiovascular disease, renal disease, and hypertension is rapidly evolving. Its important role both historically and currently in the clinical clustering phenomenon of the metabolic syndrome (MS), type 2 diabetes mellitus (T2DM), atheroscleropathy, and non-diabetic atherosclerosis is of great importance. RESULTS: Uric acid is a marker of risk and it remains controversial as to its importance as a risk factor (causative role). In this review we will attempt to justify its important role as one of the many risk factors in the development of accelerated atherosclerosis and discuss its importance of being one of the multiple injurious stimuli to the endothelium, the arterial vessel wall, and capillaries. The role of uric acid, oxidative – redox stress, reactive oxygen species, and decreased endothelial nitric oxide and endothelial dysfunction cannot be over emphasized. In the atherosclerotic prooxidative environmental milieu the original antioxidant properties of uric acid paradoxically becomes prooxidant, thus contributing to the oxidation of lipoproteins within atherosclerotic plaques, regardless of their origins in the MS, T2DM, accelerated atherosclerosis (atheroscleropathy), or non-diabetic vulnerable atherosclerotic plaques. In this milieu there exists an antioxidant – prooxidant urate redox shuttle. CONCLUSION: Elevations of uric acid > 4 mg/dl should be considered a "red flag" in those patients at risk for cardiovascular disease and should alert the clinician to strive to utilize a global risk reduction program in a team effort to reduce the complications of the atherogenic process resulting in the morbid – mortal outcomes of cardiovascular disease

Is type 2 diabetes mellitus a vascular disease (atheroscleropathy) with hyperglycemia a late manifestation? The role of NOS, NO, and redox stress.

BACKGROUND: Cardiovascular disease accounts for at least 85 percent of deaths for those patients with type 2 diabetes mellitus (T2DM). Additionally, 75 percent of these deaths are due to ischemic heart disease. HYPOTHESIS: Is type 2 diabetes mellitus a vascular disease (atheroscleropathy) with hyperglycemia a late manifestation? The role of NOS, NO, and redox stress. TESTING OF THE HYPOTHESIS: The vulnerable three arms of the eNOS reaction responsible for the generation of eNO is discussed in relation to the hypothesis: (1). The L-arginine substrate. (2). The eNOS enzyme. (3). The BH4 cofactor. IMPLICATIONS OF THE HYPOTHESIS: If we view T2DM as a vascular disease initially with a later manifestation of hyperglycemia, we may be able to better understand and modify the multiple toxicities associated with insulin resistance, metabolic syndrome, prediabetes, overt T2DM, and accelerated atherosclerosis (atheroscleropathy). The importance of endothelial nitric oxide synthase, endothelial nitric oxide, tetrahydrobiopterin (BH4), L-arginine, and redox stress are discussed in relation to endothelial cell dysfunction and the development and progression of atheroscleropathy and T2DM. In addition to the standard therapies to restore endothelial cell dysfunction and stabilization of vulnerable atherosclerotic plaques, this article will discuss the importance of folic acid (5MTHF) supplementation in this complex devastating disease process. Atheroscleropathy and hyperglycemia could be early and late manifestations, respectively, in the natural progressive history of T2DM

Wyner-Ziv Estimators: Efficient Distributed Mean Estimation with Side Information

Communication efficient distributed mean estimation is an important primitive that arises in many distributed learning and optimization scenarios such as federated learning. Without any probabilistic assumptions on the underlying data, we study the problem of distributed mean estimation where the server has access to side information. We propose \emph{Wyner-Ziv estimators}, which are communication and computationally efficient and near-optimal when an upper bound for the distance between the side information and the data is known. As a corollary, we also show that our algorithms provide efficient schemes for the classic Wyner-Ziv problem in information theory. In a different direction, when there is no knowledge assumed about the distance between side information and the data, we present an alternative Wyner-Ziv estimator that uses correlated sampling. This latter setting offers {\em universal recovery guarantees}, and perhaps will be of interest in practice when the number of users is large and keeping track of the distances between the data and the side information may not be possible