We thank the reviewers for their feedback. We will first respond to shared and then to individual comments. Additionally, reviewers 2 and 3 requested clarification regarding the advantages of DCA over other methods. For instance, one could attempt to correlate each neuron's contribution to the DCA subspace with single-neuron Studying the behavior of Kernel DCA is a direction for future studies. Additionally, we found and corrected a minor bug in Figure 1A: the SFA and DCA lines are now blue and red, respectively.

Linear dimensionality reduction methods are commonly used to extract low-dimensional structure from high-dimensional data. However, popular methods disregard temporal structure, rendering them prone to extracting noise rather than meaningful dynamics when applied to time series data. At the same time, many successful unsupervised learning methods for temporal, sequential and spatial data extract features which are predictive of their surrounding context. Combining these approaches, we introduce Dynamical Components Analysis (DCA), a linear dimensionality reduction method which discovers a subspace of high-dimensional time series data with maximal predictive information, defined as the mutual information between the past and future. We test DCA on synthetic examples and demonstrate its superior ability to extract dynamical structure compared to commonly used linear methods. We also apply DCA to several real-world datasets, showing that the dimensions extracted by DCA are more useful than those extracted by other methods for predicting future states and decoding auxiliary variables.

Motivated by the recently shown connection between self-attention and (kernel) principal component analysis (PCA), we revisit the fundamentals of PCA. Using the difference-of-convex (DC) framework, we present several novel formulations and provide new theoretical insights. In particular, we show the kernelizability and out-of-sample applicability for a PCA-like family of problems. Moreover, we uncover that simultaneous iteration, which is connected to the classical QR algorithm, is an instance of the difference-of-convex algorithm (DCA), offering an optimization perspective on this longstanding method. Further, we describe new algorithms for PCA and empirically compare them with state-of-the-art methods. Lastly, we introduce a kernelizable dual formulation for a robust variant of PCA that minimizes the $l_1$ deviation of the reconstruction errors.

Brain atlases are essential for reducing the dimensionality of neuroimaging data and enabling interpretable analysis. However, most existing atlases are predefined, group-level templates with limited flexibility and resolution. We present Deep Cluster Atlas (DCA), a graph-guided deep embedding clustering framework for generating individualized, voxel-wise brain parcellations. DCA combines a pretrained autoencoder with spatially regularized deep clustering to produce functionally coherent and spatially contiguous regions. Our method supports flexible control over resolution and anatomical scope, and generalizes to arbitrary brain structures. We further introduce a standardized benchmarking platform for atlas evaluation, using multiple large-scale fMRI datasets. Across multiple datasets and scales, DCA outperforms state-of-the-art atlases, improving functional homogeneity by 98.8% and silhouette coefficient by 29%, and achieves superior performance in downstream tasks such as autism diagnosis and cognitive decoding. We also observe that a fine-tuned pretrained model achieves superior results on the corresponding task. Codes and models are available at https://github.com/ncclab-sustech/DCA .

In classical planning and conformant planning, it is assumed that there are finitely many named objects given in advance, and only they can participate in actions and in fluents. This is the Domain Closure Assumption (DCA). However, there are practical planning problems where the set of objects changes dynamically as actions are performed; e.g., new objects can be created, old objects can be destroyed. We formulate the planning problem in first-order logic, assume an initial theory is a finite consistent set of fluent literals, discuss when this guarantees that in every situation there are only finitely many possible actions, impose a finite integer bound on the length of the plan, and propose to organize search over sequences of actions that are grounded at planning time. We show the soundness and completeness of our approach. It can be used to solve the bounded planning problems without DCA that belong to the intersection of sequential generalized planning (without sensing actions) and conformant planning, restricted to the case without the disjunction over fluent literals. We discuss a proof-of-the-concept implementation of our planner.