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Linear discriminant analysis improves class separability but struggles with non-linearly separable data. To overcome this, we introduce Deep Discriminant Analysis (DDA), which directly optimizes the Fisher criterion utilizing deep networks. To ensure stable training and avoid computational instabilities, we incorporate signed between-class variance, bound outputs with a sigmoid function, and convert multiplicative relationships into additive ones. We present two stable DDA loss functions and augment them with a probability loss, resulting in Probabilistic DDA (PDDA).

This work introduces a novel variational approach to uncalibrated photometric stereo for robustly recovering 3D surface shape, reflectance, and lighting from images under unknown illumination. Unlike classical methods that require calibration, this framework formulates the task as an unsupervised joint optimization problem for both diffuse and specular reflectance. The proposed method incorporates a specialized depth regularizer that acts separately on interior regions and object boundaries, substantially improving depth and albedo estimation.

A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate the Koopman operator without a clear connection to differential operators, the proposed method constructs finite-dimensional Koopman matrices by projecting the dynamics onto a Chebyshev basis, thereby capturing both differential and nonlinear terms.