R2: "Section 4.2 needs more clarity. It's not clear if the Fog noise model is a good model for fog corruption..."

Measuring the distance between machine-produced and human language is a critical open problem. Inspired by empirical findings from psycholinguistics on the periodicity of entropy in language, we propose FACE, a set of metrics based on Fourier Analysis of the estimated Cross-Entropy of language, for measuring the similarity between model-generated and human-written languages. Based on an open-ended generation task and the experimental data from previous studies, we find that FACE can effectively identify the human-model gap, scales with model size, reflects the outcomes of different sampling methods for decoding, correlates well with other evaluation metrics and with human judgment scores.

We thank all the reviewers for their insightful comments and suggestions. We consider our mutual information framework to be a core contribution of our paper. "forget" the simple component even when trained to completion, provided it somehow learns the simple component Thank you for pointing out the relevant papers. We plan on including a separate section with such examples in the final version. Concretely, regarding "On the spectral bias of neural networks" [1]: They consider measuring "simplicity" via the Our metrics do not suffer from this issue - they are taken with respect to the true data distribution.

Time series forecasting is critical in numerous real-world applications, requiring accurate predictions of future values based on observed patterns. While traditional forecasting techniques work well in in-domain scenarios with ample data, they struggle when data is scarce or not available at all, motivating the emergence of zero-shot and few-shot learning settings. Recent advancements often leverage large-scale foundation models for such tasks, but these methods require extensive data and compute resources, and their performance may be hindered by ineffective learning from the available training set. This raises a fundamental question: What factors influence effective learning from data in time series forecasting? Toward addressing this, we propose using Fourier analysis to investigate how models learn from synthetic and real-world time series data. Our findings reveal that forecasters commonly suffer from poor learning from data with multiple frequencies and poor generalization to unseen frequencies, which impedes their predictive performance. To alleviate these issues, we present a novel synthetic data generation framework, designed to enhance real data or replace it completely by creating task-specific frequency information, requiring only the sampling rate of the target data. Our approach, Freq-Synth, improves the robustness of both foundation as well as nonfoundation forecast models in zero-shot and few-shot settings, facilitating more reliable time series forecasting under limited data scenarios. Time series forecasting (TSF) plays a critical role in various areas, such as finance, healthcare, and energy, where accurate predictions of future values are essential for decision-making and planning. Traditionally, in-domain learning has been the common setting for developing forecasting models, where a model is trained using data from the same domain it will later be deployed in (Salinas et al., 2020; Zhou et al., 2021). This ensures that the model captures the patterns, seasonality, and trends specific to the target domain, improving its predictive performance. However, a significant challenge arises when there is scarce or no historical information available for training, limiting the ability to apply traditional in-domain learning approaches (Sarmas et al., 2022; Fong et al., 2020). In such cases, the emergence of zero-shot (ZS) and few-shot (FS) learning settings offer potential solutions. Zero-shot learning enables models to generalize to new, unseen domains without requiring domainspecific data by leveraging knowledge transfer from other domains or tasks.

Measuring the distance between machine-produced and human language is a critical open problem. Inspired by empirical findings from psycholinguistics on the periodicity of entropy in language, we propose FACE, a set of metrics based on Fourier Analysis of the estimated Cross-Entropy of language, for measuring the similarity between model-generated and human-written languages. Based on an open-ended generation task and the experimental data from previous studies, we find that FACE can effectively identify the human-model gap, scales with model size, reflects the outcomes of different sampling methods for decoding, correlates well with other evaluation metrics and with human judgment scores.

Bayesian estimation is a vital tool in robotics as it allows systems to update the belief of the robot state using incomplete information from noisy sensors. To render the state estimation problem tractable, many systems assume that the motion and measurement noise, as well as the state distribution, are all unimodal and Gaussian. However, there are numerous scenarios and systems that do not comply with these assumptions. Existing non-parametric filters that are used to model multimodal distributions have drawbacks that limit their ability to represent a diverse set of distributions. In this paper, we introduce a novel approach to nonparametric Bayesian filtering to cope with multimodal distributions using harmonic exponential distributions. This approach leverages two key insights of harmonic exponential distributions: a) the product of two distributions can be expressed as the element-wise addition of their log-likelihood Fourier coefficients, and b) the convolution of two distributions can be efficiently computed as the tensor product of their Fourier coefficients. These observations enable the development of an efficient and exact solution to the Bayes filter up to the band limit of a Fourier transform. We demonstrate our filter's superior performance compared with established nonparametric filtering methods across a range of simulated and real-world localization tasks.

Despite the notable success of language models (LMs) in various natural language processing (NLP) tasks, the reliability of LMs is susceptible to backdoor attacks. Prior research attempts to mitigate backdoor learning while training the LMs on the poisoned dataset, yet struggles against complex backdoor attacks in real-world scenarios. In this paper, we investigate the learning mechanisms of backdoor LMs in the frequency space by Fourier analysis. Our findings indicate that the backdoor mapping presented on the poisoned datasets exhibits a more discernible inclination towards lower frequency compared to clean mapping, resulting in the faster convergence of backdoor mapping. To alleviate this dilemma, we propose Multi-Scale Low-Rank Adaptation (MuScleLoRA), which deploys multiple radial scalings in the frequency space with low-rank adaptation to the target model and further aligns the gradients when updating parameters. Through downscaling in the frequency space, MuScleLoRA encourages the model to prioritize the learning of relatively high-frequency clean mapping, consequently mitigating backdoor learning. Experimental results demonstrate that MuScleLoRA outperforms baselines significantly. Notably, MuScleLoRA reduces the average success rate of diverse backdoor attacks to below 15\% across multiple datasets and generalizes to various backbone LMs, including BERT, RoBERTa, GPT2-XL, and Llama2. The codes are publicly available at https://github.com/ZrW00/MuScleLoRA.

Modeling complex systems using standard neural ordinary differential equations (NODEs) often faces some essential challenges, including high computational costs and susceptibility to local optima. To address these challenges, we propose a simulation-free framework, called Fourier NODEs (FNODEs), that effectively trains NODEs by directly matching the target vector field based on Fourier analysis. Specifically, we employ the Fourier analysis to estimate temporal and potential high-order spatial gradients from noisy observational data. We then incorporate the estimated spatial gradients as additional inputs to a neural network. Furthermore, we utilize the estimated temporal gradient as the optimization objective for the output of the neural network. Later, the trained neural network generates more data points through an ODE solver without participating in the computational graph, facilitating more accurate estimations of gradients based on Fourier analysis. These two steps form a positive feedback loop, enabling accurate dynamics modeling in our framework. Consequently, our approach outperforms state-of-the-art methods in terms of training time, dynamics prediction, and robustness. Finally, we demonstrate the superior performance of our framework using a number of representative complex systems.