Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms.

B we provide detailed proofs of the results presented in Sec. 3 .

Time series classification using deep neural networks, such as convolutional neural networks (CNN), operate on the spectral decomposition of the time series computed using a preprocessing step. This step can include a large number of hyperparameters, such as window length, filter widths, and filter shapes, each with a range of possible values that must be chosen using time and data intensive cross-validation procedures. We propose the wavelet deconvolution (WD) layer as an efficient alternative to this preprocessing step that eliminates a significant number of hyperparameters. The WD layer uses wavelet functions with adjustable scale parameters to learn the spectral decomposition directly from the signal. Using backpropagation, we show the scale parameters can be optimized with gradient descent. Furthermore, the WD layer adds interpretability to the learned time series classifier by exploiting the properties of the wavelet transform.

Non-linear spectral decompositions of images based on one-homogeneous functionals such as total variation have gained considerable attention in the last few years. Due to their ability to extract spectral components corresponding to objects of different size and contrast, such decompositions enable filtering, feature transfer, image fusion and other applications. However, obtaining this decomposition involves solving multiple non-smooth optimisation problems and is therefore computationally highly intensive. In this paper, we present a neural network approximation of a non-linear spectral decomposition. We report up to four orders of magnitude ( 10,000) speedup in processing of mega-pixel size images, compared to classical GPU implementations. Our proposed network, TVspecNET, is able to implicitly learn the underlying PDE and, despite being entirely data driven, inherits invariances of the model based transform. To the best of our knowledge, this is the first approach towards learning a non-linear spectral decomposition of images. Not only do we gain a staggering computational advantage, but this approach can also be seen as a step towards studying neural networks that can decompose an image into spectral components defined by a user rather than a handcrafted functional.

Knowledge distillation is an effective and hardware-friendly method, which plays a key role in lightweighting remote sensing object detection. However, existing distillation methods often encounter the issue of mixed features in remote sensing images (RSIs), and neglect the discrepancies caused by subtle feature variations, leading to entangled knowledge confusion. To address these challenges, we propose an architecture-agnostic distillation method named Dual-Stream Spectral Decoupling Distillation (DS2D2) for universal remote sensing object detection tasks. Specifically, DS2D2 integrates explicit and implicit distillation grounded in spectral decomposition. Firstly, the first-order wavelet transform is applied for spectral decomposition to preserve the critical spatial characteristics of RSIs. Leveraging this spatial preservation, a Density-Independent Scale Weight (DISW) is designed to address the challenges of dense and small object detection common in RSIs. Secondly, we show implicit knowledge hidden in subtle student-teacher feature discrepancies, which significantly influence predictions when activated by detection heads. This implicit knowledge is extracted via full-frequency and high-frequency amplifiers, which map feature differences to prediction deviations. Extensive experiments on DIOR and DOTA datasets validate the effectiveness of the proposed method. Specifically, on DIOR dataset, DS2D2 achieves improvements of 4.2% in AP50 for RetinaNet and 3.8% in AP50 for Faster R-CNN, outperforming existing distillation approaches. The source code will be available at https://github.com/PolarAid/DS2D2.

Time series classification using deep neural networks, such as convolutional neural networks (CNN), operate on the spectral decomposition of the time series computed using a preprocessing step. This step can include a large number of hyperparameters, such as window length, filter widths, and filter shapes, each with a range of possible values that must be chosen using time and data intensive cross-validation procedures. We propose the wavelet deconvolution (WD) layer as an efficient alternative to this preprocessing step that eliminates a significant number of hyperparameters. The WD layer uses wavelet functions with adjustable scale parameters to learn the spectral decomposition directly from the signal. Using backpropagation, we show the scale parameters can be optimized with gradient descent. Furthermore, the WD layer adds interpretability to the learned time series classifier by exploiting the properties of the wavelet transform.

Time series classification using deep neural networks, such as convolutional neural networks (CNN), operate on the spectral decomposition of the time series computed using a preprocessing step.

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems.

The network is a differentiable deep architecture consisting of feature extraction and diffusion distance modules for computing diffusion distance on image by end-to-end training. We design low resolution kernel matching loss and high resolution segment matching loss to enforce the network's output to be consistent with human-labeled image segments.

Time series classification using deep neural networks, such as convolutional neural networks (CNN), operate on the spectral decomposition of the time series computed using a preprocessing step.