Exponential Family Matrix Completion under Structural Constraints

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low--rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin--tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low--rank, such as block--sparsity, or a superposition structure of low--rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of exponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized $M$--estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.Comment: 20 pages, 9 figure

Effects of alpha, beta momorcharin fruit extract with the combination of paclitaxel in the treatment of glioma cancer in-vivo.

The vegetable Momordica charantia L., (family: Cucurbitaceae) is a scientific name of the plant and its fruit. It is also known by other names, for instance in the USA it is known as Bitter gourd or balsam pear while its referred to as the African cucumber in many African countries.  M.charantiais believed to posse’s anti-carcinogenic properties and it can modulate its effect via xenobiotic metabolism and oxidative stress. This study was specifically designed to investigate the cellular mechanisms whereby α, β momorcharin an extract of M. charantiacan induce cell death with the combination of paclitaxel.  Different concentration (200µM - 1000µM)of the α, β momorcharin fruit extract were treated (24 hrs incubation) separately with three different cancer cell lines 1321N1, Gos-3, U87-MG and normal L6 muscle cell line. The results also show that paclitaxel(250 µg) with (1000 µM) of the α, β momorcharinextract of M. Charantia,and result in significant decreases in cell viability for each cell line, these effects were additive compared to the individual effect of paclitaxel.   &nbsp

Effects of α, β momorcharin extract of momordica charantia in intracellular free calcium on cancer cell lines.

The Momordica charantia L., (family: Cucurbitaceae) is a scientific name of the plant and its fruit. It is also known by other names, for instance in the USA it is known as Bitter gourd or balsam pear while its referred to as the African cucumber in many African countries. This study was specifically designed to investigate the cellular mechanisms whereby alpha, beta momorcharin an extract of M. charantiacan induce cell death measuring the elevation in intracellular free calcium concentrations in three different cancer cell lines 1321N1, Gos-3 and U-87. The results show that incubation of the three cancer cell lines 1321N1, Gos-3 and U-87 with  α, β momorcharin can result in significant (p < 0.05) time-dependent increases in [Ca2+]i in all three cancer cell lines compared to control (untreated) cells. Maximal increases in [Ca2+]i was attained after 420 min of incubation.In control (untreated cell lines), [Ca2+]i remained more or less stable in both cell lines after 420 min. The results also show that the increase in [Ca2+]i in Gos-3 cell line was much more pronounced following incubation with α, β momorcharin compared to 1321N1 and U-87 cell line. The results show that incubation of the three cancer cell lines with  momorcharin can result in significant (p < 0.05) time-dependent increases in [Ca2+]i in all three cancer cell lines compared to control (untreated) cells. Maximal increases in [Ca2+]i was attained after 420 min of incubation. In control (untreated cell lines), [Ca2+]i remained more or less stable in all three cell lines after 420 min. These results clearly show that  α, β momorcharin extract of M. charantia is exerting its anti- cancer effect via an insult to the mitochondria resulting in apoptosis, calcium overloading and subsequently, cell death

Effects of alpha, beta momorcharin fruit extract with the combination of paclitaxel in the treatment of glioma cancer in-vivo.

The vegetable Momordica charantia L., (family: Cucurbitaceae) is a scientific name of the plant and its fruit. It is also known by other names, for instance in the USA it is known as Bitter gourd or balsam pear while its referred to as the African cucumber in many African countries.  M.charantiais believed to posse’s anti-carcinogenic properties and it can modulate its effect via xenobiotic metabolism and oxidative stress. This study was specifically designed to investigate the cellular mechanisms whereby α, β momorcharin an extract of M. charantiacan induce cell death with the combination of paclitaxel.  Different concentration (200µM - 1000µM)of the α, β momorcharin fruit extract were treated (24 hrs incubation) separately with three different cancer cell lines 1321N1, Gos-3, U87-MG and normal L6 muscle cell line. The results also show that paclitaxel(250 µg) with (1000 µM) of the α, β momorcharinextract of M. Charantia,and result in significant decreases in cell viability for each cell line, these effects were additive compared to the individual effect of paclitaxel.   &nbsp

Quantitative dynamic cellular imaging based on 3d unwrapped optical coherence phase microscopy

Phase wrapping artifacts are frequently encountered in phase resolved imaging and sensing techniques based on interferometry. When the value of the actual phase (φ) ranges beyond (-π, π], the extracted phase value (wrapped phase Ψ) is artificially increased or decreased by a multiple of 2π. In this study, we develop a 3D phase unwrapping method that exploits the correlation of 3D phase data (φ(x,y,t)) over the other dimension (t). We validate our 3D unwrapping method using simulated data, as well as experimental data obtained by optically computed phase microscopy (OCPM) recently developed in our lab. Our results show that the 3D phase unwrapping method achieves accurate and robust reconstruction of phase φ(x,y,t). The application of the 3D phase unwrapping method is not limited to OCPM. Instead, it is a generic approach that can be used for other phase imaging modalities

Effects of α, β momorcharin extract of momordica charantia in intracellular free calcium on cancer cell lines.

The Momordica charantia L., (family: Cucurbitaceae) is a scientific name of the plant and its fruit. It is also known by other names, for instance in the USA it is known as Bitter gourd or balsam pear while its referred to as the African cucumber in many African countries. This study was specifically designed to investigate the cellular mechanisms whereby alpha, beta momorcharin an extract of M. charantiacan induce cell death measuring the elevation in intracellular free calcium concentrations in three different cancer cell lines 1321N1, Gos-3 and U-87. The results show that incubation of the three cancer cell lines 1321N1, Gos-3 and U-87 with  α, β momorcharin can result in significant (p < 0.05) time-dependent increases in [Ca2+]i in all three cancer cell lines compared to control (untreated) cells. Maximal increases in [Ca2+]i was attained after 420 min of incubation.In control (untreated cell lines), [Ca2+]i remained more or less stable in both cell lines after 420 min. The results also show that the increase in [Ca2+]i in Gos-3 cell line was much more pronounced following incubation with α, β momorcharin compared to 1321N1 and U-87 cell line. The results show that incubation of the three cancer cell lines with  momorcharin can result in significant (p < 0.05) time-dependent increases in [Ca2+]i in all three cancer cell lines compared to control (untreated) cells. Maximal increases in [Ca2+]i was attained after 420 min of incubation. In control (untreated cell lines), [Ca2+]i remained more or less stable in all three cell lines after 420 min. These results clearly show that  α, β momorcharin extract of M. charantia is exerting its anti- cancer effect via an insult to the mitochondria resulting in apoptosis, calcium overloading and subsequently, cell death

The Implicit Bias of Gradient Descent on Separable Data

We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods.Comment: Final JMLR version, with improved discussions over v3. Main improvements in journal version over conference version (v2 appeared in ICLR): We proved the measure zero case for main theorem (with implications for the rates), and the multi-class cas