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In this paper, we consider a problem of estimating a large loss probability of financial derivatives portfolio, which are commonly modeled as nested expectations. However, the cost of nested simulation may be too expensive and thus multilevel Monte Carlo (MLMC) method is recently used to reduce the nested simulation complexity. When using antithetic MLMC to solve the indicator function, we get the complexity of O(ϵ -5/2). To decrease the computational burden, we use a Fourier transform method to modify the form of indicator function. The new estimator is sufficiently smooth and enables the antithetic MLMC method to achieve a better complexity. In addtion, we combine quasi-Monte Carlo (QMC) with MLMC to reduce the variance of inner estimator. Numerical results show that using the Fourier transform method in both MLMC and MLQMC can attain the optimal complexity O(ϵ -2). [ABSTRACT FROM AUTHOR]

Real-world data often exhibits a long-tailed distribution in practical scenarios. However, deep learning models usually face challenges when it comes to effectively identifying infrequent classes amidst the abundance of prevalent ones. The fundamental issue lies in the scarcity of available information for tail classes. A highly intuitive approach is to uncover a greater amount of valuable information specifically tailored to these tail classes. We find that textual information of class names and frequency domain information of images are ignored by previous works in long-tailed visual recognition. Therefore, we propose a Text-Guided Fourier Augmentation (TGFA) method with the aid of language models and the Fourier transform to excavate more useful information for tail classes. Extensive experiments demonstrate that our proposed method effectively enriches training data on-the-fly, allowing for an end-to-end one-stage supervised contrastive learning framework that surpasses other methods including two-stage or multi-experts methods, in terms of efficiency and performance. • A novel Text-Guided Fourier Augmentation method for long-tailed visual recognition. • First work fusing textual and Fourier information to enhancing long-tailed learning. • Our end-to-end method outperforms two-stage or multi-experts long-tailed methods. • Superior over other methods in terms of efficiency and performance. [ABSTRACT FROM AUTHOR]

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]

The work is devoted to integro-differential equations related to stochastic processes. We study the relationship between differential equations with random perturbations — stochastic differential equations (SDEs) — and deterministic equations for the probability characteristics of processes determined by random perturbations. The resulting deterministic pseudodifferential equations are investigated by semigroup methods and Fourier transform methods. [ABSTRACT FROM AUTHOR]

Although great success has been achieved in various computer vision tasks, deep neural networks (DNNs) suffer dramatic performance degradation when evaluated on out-of-distribution data. Domain generalization (DG) is proposed to handle this problem by learning domain-agnostic information from multiple source domains to generalize well on unseen target domains. Several methods resort to Fourier transform due to its simplicity and efficiency. They argue that amplitude spectra imply domain-specific information, which should be suppressed, while phase counterparts imply domain-agnostic information, which should be preserved. However, these methods only suppress the domain-specific information in source domains and neglect the relationship with target domains, leading to the persistence of the domain gap. Besides, these methods preserve domain-agnostic information by keeping phase components unchanged, causing them to be underutilized. In this paper, we propose Dual Branch Augmentation Module (DBAM) by leveraging Fourier transform and taking advantage of both amplitude and phase spectra. For the amplitude branch, we propose Inner-domain Amplitude Distribution Rectification (IADR) and Cross-domain Amplitude Dirichlet Mixup (CADM) to stabilize the training process and explore more feature space. In addition, we propose Test-time Amplitude Prototype Calibration (TAPC) to construct the connection between source and target domains during evaluation to further mitigate the domain gap. For the phase branch, we propose Random Symmetric Phase Perturbation (RSPP) to enhance the robustness for recognizing domain-agnostic information. With the combined contributions of the two branches, DBAM significantly surpasses other state-of-the-art (SOTA) methods. Extensive experiments on four benchmarks and further analysis demonstrate the effectiveness of DBAM. [ABSTRACT FROM AUTHOR]