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Computing an element of the Clarke subdifferential of a function represented by a program is an important problem in modern non-smooth optimization. Existing algorithms either are computationally inefficient in the sense that the computational cost depends on the input dimension or can only cover simple programs such as polynomial functions with branches. In this work, we show that a generalization of the latter algorithm can efficiently compute an element of the Clarke subdifferential for programs consisting of analytic functions and linear branches, which can represent various non-smooth functions such as max, absolute values, and piecewise analytic functions with linear boundaries, as well as any program consisting of these functions such as neural networks with non-smooth activation functions. Our algorithm first finds a sequence of branches used for computing the function value at a random perturbation of the input; then, it returns an element of the Clarke subdifferential by running the backward pass of the reverse-mode automatic differentiation following those branches. The computational cost of our algorithm is at most that of the function evaluation multiplied by some constant independent of the input dimension n, if a program consists of piecewise analytic functions defined by linear branches, whose arities and maximum depths of branches are independent of n.

This study was conducted to investigate the developmental characteristics of adipocytes and to identify selectively white and brown adipocytes through Fourier transform infrared (FTIR) spectroscopy. For the developmental characterization of adipocytes, cells and conditioned media of white and brown adipocytes were respectively collected and analyzed. A higher amide I/amide II ratio was observed in the conditioned medium of brown adipocyte than in that of white adipocyte, indicating differences in secretory protein profiles. In contrast, an amide I/amide II ratio was higher in white adipocytes than in brown adipocytes, and mature adipocytes have higher lipid amounts than pre-adipocytes. Lipid acyl chain length was the longest in white adipocytes. These differences suggested that FTIR spectroscopy can be used to characterize developmental stages and/or types of adipocytes. To identify the possibility of selectively classifying adipose-derived stem cells, FTIR spectroscopy spectra were obtained in cells before/after white/brown adipocyte differentiation using FTIR spectroscopy and then analyzed by the principal component analysis method. All data indicated that the discrimination between adipocytes was possible in the analysis of the infrared spectroscopy spectrum by the principal component analysis technique. This study suggested the possibility of FTIR spectroscopy as a new type of cell sorting system without tagging.