While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.

HINT consists of several sequentially arranged lightweight interpolation blocks.

Multivariate time series in domains such as finance, climate science, and healthcare often exhibit long-term trends, seasonal patterns, and short-term fluctuations, complicating causal inference under non-stationarity and autocorrelation. Existing causal discovery methods typically operate on raw observations, making them vulnerable to spurious edges and misattributed temporal dependencies. We introduce a decomposition-based causal discovery framework that separates each time series into trend, seasonal, and residual components and performs component-specific causal analysis. Trend components are assessed using stationarity tests, seasonal components using kernel-based dependence measures, and residual components using constraint-based causal discovery. The resulting component-level graphs are integrated into a unified multi-scale causal structure. This approach isolates long- and short-range causal effects, reduces spurious associations, and improves interpretability. Across extensive synthetic benchmarks and real-world climate data, our framework more accurately recovers ground-truth causal structure than state-of-the-art baselines, particularly under strong non-stationarity and temporal autocorrelation.

Accurate interpolation algorithms are highly desired in various theoretical and engineering scenarios. Unlike the traditional numerical algorithms that have exact zero-residual constraints on observed points, the neural network-based interpolation methods exhibit non-zero residuals at these points. These residuals, which provide observations of an underlying residual function, can guide predicting interpolation functions, but have not been exploited by the existing approaches. To fill this gap, we propose Hierarchical INTerpolation Network (HINT), which utilizes the residuals on observed points to guide target function estimation in a hierarchical fashion. HINT consists of several sequentially arranged lightweight interpolation blocks. The first interpolation block estimates the main component of the target function, while subsequent blocks predict the residual components using observed points residuals of the preceding blocks. The main component and residual components are accumulated to form the final interpolation results. Furthermore, under the assumption that finer residual prediction requires a more focused attention range on observed points, we utilize hierarchical local constraints in correlation modeling between observed and target points. Extensive experiments demonstrate that HINT outperforms existing interpolation algorithms significantly in terms of interpolation accuracy across a wide variety of datasets, which underscores its potential for practical scenarios.

Accurate traffic forecasting plays a vital role in intelligent transportation systems, enabling applications such as congestion control, route planning, and urban mobility optimization. However, traffic forecasting remains challenging due to two key factors: (1) complex spatial dependencies arising from dynamic interactions between road segments and traffic sensors across the network, and (2) the coexistence of multi-scale periodic patterns (e.g., daily and weekly periodic patterns driven by human routines) with irregular fluctuations caused by unpredictable events (e.g., accidents, weather, or construction). To tackle these challenges, we propose HyperD (Hybrid Periodic Decoupling), a novel framework that decouples traffic data into periodic and residual components. The periodic component is handled by the Hybrid Periodic Representation Module, which extracts fine-grained daily and weekly patterns using learnable periodic embeddings and spatial-temporal attention. The residual component, which captures non-periodic, high-frequency fluctuations, is modeled by the Frequency-Aware Residual Representation Module, leveraging complex-valued MLP in frequency domain. To enforce semantic separation between the two components, we further introduce a Dual-View Alignment Loss, which aligns low-frequency information with the periodic branch and high-frequency information with the residual branch. Extensive experiments on four real-world traffic datasets demonstrate that HyperD achieves state-of-the-art prediction accuracy, while offering superior robustness under disturbances and improved computational efficiency compared to existing methods.

Deterministically controlling the target generation language of large multilingual language models (LLMs) remains a fundamental challenge, particularly in zero-shot settings where neither explicit language prompts nor fine-tuning are available. In this work, we investigate whether sparse autoencoder (SAE) features, previously shown to correlate with interpretable model behaviors, can be leveraged to steer the generated language of LLMs during inference. Leveraging pretrained SAEs on the residual streams of Gemma-2B and Gemma-9B, we identify features whose activations differ most significantly between English and four target languages: Chinese, Japanese, Spanish, and French. By modifying just a single SAE feature at one transformer layer, we achieve controlled language shifts with up to 90\% success, as measured by FastText language classification, while preserving semantic fidelity according to LaBSE (Language-Agnostic BERT Sentence Embedding) similarity. Our analysis reveals that language steering is most effective in mid-to-late transformer layers and is amplified by specific attention heads disproportionately associated with language-sensitive SAE features. These results demonstrate the promise of sparse feature steering as a lightweight and interpretable mechanism for controllable multilingual generation.

Conformal prediction offers finite-sample coverage guarantees under minimal assumptions. However, existing methods treat the entire modeling process as a black box, overlooking opportunities to exploit modular structure. We introduce a conformal prediction framework for two-stage sequential models, where an upstream predictor generates intermediate representations for a downstream model. By decomposing the overall prediction residual into stage-specific components, our method enables practitioners to attribute uncertainty to specific pipeline stages. We develop a risk-controlled parameter selection procedure using family-wise error rate (FWER) control to calibrate stage-wise scaling parameters, and propose an adaptive extension for non-stationary settings that preserves long-run coverage guarantees. Experiments on synthetic distribution shifts, as well as real-world supply chain and stock market data, demonstrate that our approach maintains coverage under conditions that degrade standard conformal methods, while providing interpretable stage-wise uncertainty attribution. This framework offers diagnostic advantages and robust coverage that standard conformal methods lack.

We present StructuralDecompose, an R package for modular and interpretable time series decomposition. Unlike existing approaches that treat decomposition as a monolithic process, StructuralDecompose separates the analysis into distinct components: changepoint detection, anomaly detection, smoothing, and decomposition. This design provides flexibility and robust- ness, allowing users to tailor methods to specific time series characteristics. We demonstrate the package on simulated and real-world datasets, benchmark its performance against state-of-the- art tools such as Rbeast and autostsm, and discuss its role in interpretable machine learning workflows.

Abstract:In the metro intelligent transportation system, accurate transfer passenger flow prediction is a key link in optimizing operation plans and improving transportation efficiency. To further improve the theory of metro internal transfer passenger flow prediction and provide more reliable support for intelligent operation decisions, this paper innovatively proposes a metro transfer passenger flow prediction model that integrates the Seasonal and Trend decomposition using Loess (STL) method and Gated Recurrent Unit (GRU).In practical application, the model first relies on the deep learning library Keras to complete the construction and training of the GRU model, laying the foundation for subsequent prediction; then preprocesses the original metro card swiping data, uses the graph-based depth-first search algorithm to identify passengers' travel paths, and further constructs the transfer passenger flow time series; subsequently adopts the STL time series decomposition algorithm to decompose the constructed transfer p assenger flow time series into trend component, periodic component and residual component, and uses the 3σ principle to eliminate and fill the outliers in the residual component, and finally completes the transfer passenger flow prediction.Taking the trans fer passenger flow data of a certain metro station as the research sample, the validity of the model is verified. The results show that compared with Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), and the combined model of STL time series decom position method and Long Short-Term Memory (STL-LSTM), the STL-GRU combined prediction model significantly improves the prediction accuracy of transfer passenger flow on weekdays (excluding Fridays), Fridays and rest days, with the mean absolute percentage error (MAPE) of the prediction results reduced by at least 2.3, 1.36 and 6.42 percentage points respectively. This study focuses on the field of metro transfer passenger flow prediction, aiming to break through existing technical bottlenecks through the construction of an innovative model and provide more accurate decision -making basis for metro operation management. Transfer stations, as passenger flow distribution hubs, their flow dynamics are directly related to operational efficiency and service quality.