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We consider a model for logistic regression where only a subset of features of size $p$ is used for training a linear classifier over $n$ training samples. The classifier is obtained by running gradient descent on logistic loss. For this model, we investigate the dependence of the classification error on the ratio $\kappa =p/n$. First, building on known deterministic results on the implicit bias of gradient descent, we uncover a phase-transition phenomenon for the case of Gaussian features: the classification error of the gradient descent solution is the same as that of the maximum-likelihood solution when $\kappa <\kappa _\star $, and that of the support vector machine when $\kappa>\kappa _\star $, where $\kappa _\star $ is a phase-transition threshold. Next, using the convex Gaussian min–max theorem, we sharply characterize the performance of both the maximum-likelihood and the support vector machine solutions. Combining these results, we obtain curves that explicitly characterize the classification error for varying values of $\kappa $. The numerical results validate the theoretical predictions and unveil double-descent phenomena that complement similar recent findings in linear regression settings as well as empirical observations in more complex learning scenarios.

We report a systematic study on the structural and electronic properties of Bi2Te3−xSex topological insulator alloy grown by molecular beam epitaxy (MBE). A mixing ratio of Bi2Se3 to Bi2Te3 was controlled by varying the Bi:Te:Se flux ratio. X-ray diffraction and Raman spectroscopy measurements indicate the high crystalline quality for the as-grown Bi2Te3−xSex films. Substitution of Te by Se is also revealed from both analyses. The surfaces of the films exhibit terrace-like quintuple layers and their size of the characteristic triangular terraces decreases monotonically with increasing Se content. However, the triangular terrace structure gradually recovers as the Se content further increases. Most importantly, the angle-resolved photoemission spectroscopy results provide evidence of single-Dirac-cone like surface states in which Bi2Te3−xSex with Se/Te-substitution leads to tunable surface states. Our results demonstrate that by fine-tuned MBE growth conditions, Bi2Te3−xSex thin film alloys with tunable topological surface states can be obtained, providing an excellent platform for exploring the novel device applications based on this compound.

We consider a model for logistic regression where only a subset of features of size p is used for training a linear classifier over n training samples. The classifier is obtained by running gradient-descent (GD) on the logisticloss. For this model, we investigate the dependence of the generalization error on the overparameterization ratio κ = p/n. First, building on known deterministic results on convergence properties of the GD, we uncover a phase-transition phenomenon for the case of Gaussian regressors: the generalization error of GD is the same as that of the maximum-likelihood (ML) solution when κ < κ ⋆ , and that of the max-margin (SVM) solution when κ > κ ⋆ . Next, using the convex Gaussian minmax theorem (CGMT), we sharply characterize the performance of both the ML and SVM solutions. Combining these results, we obtain curves that explicitly characterize the generalization error of GD for varying values of κ. The numerical results validate the theoretical predictions and unveil "double-descent" phenomena that complement similar recent observations in linear regression settings.

A fundamental understanding of the charge transport mechanism in two-dimensional semiconductors (e.g., MoS 2) is crucial for fully exploring their potential in electronic and optoelectronic devices. By using monolayer graphene as the barrier-free contact to MoS 2 , we show that the field-modulated conductivity can be used to probe the electronic structure of the localized states. A series of regularly distributed plateaus were observed in the gate-dependent transfer curves. Calculations based on the variable-range hopping theory indicate that such plateaus can be attributed to the discrete localized states near mobility edge. This method provides an effective approach to directly profiling the localized states in conduction channel with an ultrahigh resolution up to 1 meV.