Asymmetric matrix sensing by gradient descent with small random initialization

We study matrix sensing, which is the problem of reconstructing a low-rank matrix from a few linear measurements. It can be formulated as an overparameterized regression problem, which can be solved by factorized gradient descent when starting from a small random initialization. Linear neural networks, and in particular matrix sensing by factorized gradient descent, serve as prototypical models of non-convex problems in modern machine learning, where complex phenomena can be disentangled and studied in detail. Much research has been devoted to studying special cases of asymmetric matrix sensing, such as asymmetric matrix factorization and symmetric positive semi-definite matrix sensing. Our key contribution is introducing a continuous differential equation that we call the $\textit{perturbed gradient flow}$. We prove that the perturbed gradient flow converges quickly to the true target matrix whenever the perturbation is sufficiently bounded. The dynamics of gradient descent for matrix sensing can be reduced to this formulation, yielding a novel proof of asymmetric matrix sensing with factorized gradient descent. Compared to directly analyzing the dynamics of gradient descent, the continuous formulation allows bounding key quantities by considering their derivatives, often simplifying the proofs. We believe the general proof technique may prove useful in other settings as well

Unexpected Scaling of the Performance of Carbon Nanotube Transistors

We show that carbon nanotube transistors exhibit scaling that is qualitatively different than conventional transistors. The performance depends in an unexpected way on both the thickness and the dielectric constant of the gate oxide. Experimental measurements and theoretical calculations provide a consistent understanding of the scaling, which reflects the very different device physics of a Schottky barrier transistor with a quasi-one-dimensional channel contacting a sharp edge. A simple analytic model gives explicit scaling expressions for key device parameters such as subthreshold slope, turn-on voltage, and transconductance.Comment: 4 pages, 4 figure

Multidrug Resistance in Breast Cancer: From In Vitro Models to Clinical Studies

The development of multidrug resistance (MDR) and subsequent relapse on therapy is a widespread problem in breast cancer, but our understanding of the underlying molecular mechanisms is incomplete. Numerous studies have aimed to establish the role of drug transporter pumps in MDR and to link their expression to response to chemotherapy. The ATP-binding cassette (ABC) transporters are central to breast cancer MDR, and increases in ABC expression levels have been shown to correlate with decreases in response to various chemotherapy drugs and a reduction in overall survival. But as there is a large degree of redundancy between different ABC transporters, this correlation has not been seen in all studies. This paper provides an introduction to the key molecules associated with breast cancer MDR and summarises evidence of their potential roles reported from model systems and clinical studies. We provide possible explanations for why despite several decades of research, the precise role of ABC transporters in breast cancer MDR remains elusive

MODIS Cloud Optical Property Retrieval Uncertainties Derived from Pixel-Level VNIR/SWIR Radiometric Uncertainties

Moderate Resolution Imaging Spectroradiometer (MODIS) retrievals of optical thickness and effective particle radius for liquid water and ice phase clouds employ a well-known VNIR/ SWIR solar reflectance technique. For this type of algorithm, we evaluate the quantitative uncertainty in simultaneous retrievals of these two cloud parameters to pixel-level radiometric calibration estimates and other fundamental (and tractable) error sources

The Use of Interacting Marketing Models as Framework for Research

The recent emphasis on consumers as the raison d\u27etre of the business firm has led marketers to neglect or deemphasize the roles of the other participants in the marketing system. Yet, an understanding of the marketing system requires the development of a comprehensive marketing model focusing on the consumers, distributive institutions, manufacturers, and other participants involved in the marketing of the given product(s)

Single-Walled Carbon Nanotubes as Shadow Masks for Nanogap Fabrication

We describe a technique for fabricating nanometer-scale gaps in Pt wires on insulating substrates, using individual single-walled carbon nanotubes as shadow masks during metal deposition. More than 80% of the devices display current-voltage dependencies characteristic of direct electron tunneling. Fits to the current-voltage data yield gap widths in the 0.8-2.3 nm range for these devices, dimensions that are well suited for single-molecule transport measurements

Diffusion of New Products in Heterogeneous Populations: Incorporating Stochastic Coefficients

Diffusion models have had a major impact on the literature and practice of marketing science. Following the pioneering work of Bass (1969), which suggested a deterministic model for homogeneous populations, the basic diffusion model has been extended to incorporate: changes in the market potential over time (Mahajan & Peterson 1978); complimentarity, substitutability, contigent & independent relations of the new product with other brands in the market place (Peterson & Mahajan 1978); spatial diffusion pattern (Mahajan & Peterson 1979); varying word-of-mouth effects (Easingwood, Mahajan & Muller 1983); various marketing mix effects including the effect of price on both innovation and imitation coefficients (Robinson and Lakhani 1975) or advertising effect on the innovation coefficient (Horsky and Simon 1983). competitive effects (Eliashberg & Jeuland 1982, Fershtman, Mahajan and Muller 1983

Implicit regularization in AI meets generalized hardness of approximation in optimization -- Sharp results for diagonal linear networks

Understanding the implicit regularization imposed by neural network architectures and gradient based optimization methods is a key challenge in deep learning and AI. In this work we provide sharp results for the implicit regularization imposed by the gradient flow of Diagonal Linear Networks (DLNs) in the over-parameterized regression setting and, potentially surprisingly, link this to the phenomenon of phase transitions in generalized hardness of approximation (GHA). GHA generalizes the phenomenon of hardness of approximation from computer science to, among others, continuous and robust optimization. It is well-known that the $\ell^1$-norm of the gradient flow of DLNs with tiny initialization converges to the objective function of basis pursuit. We improve upon these results by showing that the gradient flow of DLNs with tiny initialization approximates minimizers of the basis pursuit optimization problem (as opposed to just the objective function), and we obtain new and sharp convergence bounds w.r.t.\ the initialization size. Non-sharpness of our results would imply that the GHA phenomenon would not occur for the basis pursuit optimization problem -- which is a contradiction -- thus implying sharpness. Moreover, we characterize $\textit{which}$ $\ell_1$ minimizer of the basis pursuit problem is chosen by the gradient flow whenever the minimizer is not unique. Interestingly, this depends on the depth of the DLN

Spin dependent transport of ``nonmagnetic metal/zigzag nanotube encapsulating magnetic atoms/nonmagnetic metal'' junctions

Towards a novel magnetoresistance (MR) device with a carbon nanotube, we propose ``nonmagnetic metal/zigzag nanotube encapsulating magnetic atoms/nonmagnetic metal'' junctions. We theoretically investigate how spin-polarized edges of the nanotube and the encapsulated magnetic atoms influence on transport. When the on-site Coulomb energy divided by the magnitude of transfer integral, $U/|t|$, is larger than 0.8, large MR effect due to the direction of spins of magnetic atoms, which has the magnitude of the MR ratio of about 100%, appears reflecting such spin-polarized edges.Comment: 4 pages, 3 figures, accepted for publication in Synth. Metal