In this paper, we implement kSubS using a transformer-based subgoal module coupled with the classical best-first search framework.

To obtain excellent deep neural architectures, a series of techniques are carefully designed in EfficientNets. The giant formula for simultaneously enlarging the resolution, depth and width provides us a Rubik's cube for neural networks. So that we can find networks with high efficiency and excellent performance by twisting the three dimensions. This paper aims to explore the twisting rules for obtaining deep neural networks with minimum model sizes and computational costs. Different from the network enlarging, we observe that resolution and depth are more important than width for tiny networks. Therefore, the original method, \ie the compound scaling in EfficientNet is no longer suitable. To this end, we summarize a tiny formula for downsizing neural architectures through a series of smaller models derived from the EfficientNet-B0 with the FLOPs constraint. Experimental results on the ImageNet benchmark illustrate that our TinyNet performs much better than the smaller version of EfficientNets using the inversed giant formula. For instance, our TinyNet-E achieves a 59.9\% Top-1 accuracy with only 24M FLOPs, which is about 1.9\% higher than that of the previous best MobileNetV3 with similar computational cost.

Image restoration techniques, spanning from the convolution to the transformer paradigm, have demonstrated robust spatial representation capabilities to deliver high-quality performance.Yet, many of these methods, such as convolution and the Feed Forward Network (FFN) structure of transformers, primarily leverage the basic first-order channel interactions and have not maximized the potential benefits of higher-order modeling. To address this limitation, our research dives into understanding relationships within the channel dimension and introduces a simple yet efficient, high-order channel-wise operator tailored for image restoration. Instead of merely mimicking high-order spatial interaction, our approach offers several added benefits: Efficiency: It adheres to the zero-FLOP and zero-parameter principle, using a spatial-shifting mechanism across channel-wise groups. Simplicity: It turns the favorable channel interaction and aggregation capabilities into element-wise multiplications and convolution units with $1 \times 1$ kernel. Our new formulation expands the first-order channel-wise interactions seen in previous works to arbitrary high orders, generating a hierarchical receptive field akin to a Rubik's cube through the combined action of shifting and interactions. Furthermore, our proposed Rubik's cube convolution is a flexible operator that can be incorporated into existing image restoration networks, serving as a drop-in replacement for the standard convolution unit with fewer parameters overhead. We conducted experiments across various low-level vision tasks, including image denoising, low-light image enhancement, guided image super-resolution, and image de-blurring. The results consistently demonstrate that our Rubik's cube operator enhances performance across all tasks.

Progress in understanding expert performance is limited by the scarcity of quantitative data on long-term knowledge acquisition and deployment. Here we use the Rubik's Cube as a cognitive model system existing at the intersection of puzzle solving, skill learning, expert knowledge, cultural transmission, and group theory. By studying competitive cube communities, we find evidence for universality in the collective learning of the Rubik's Cube in both sighted and blindfolded conditions: expert performance follows exponential progress curves whose parameters reflect the delayed acquisition of algorithms that shorten solution paths. Blindfold solves form a distinct problem class from sighted solves and are constrained not only by expert knowledge but also by the skill improvements required to overcome short-term memory bottlenecks, a constraint shared with blindfold chess. Cognitive artifacts such as the Rubik's Cube help solvers navigate an otherwise enormous mathematical state space. In doing so, they sustain collective intelligence by integrating communal knowledge stores with individual expertise and skill, illustrating how expertise can, in practice, continue to deepen over the course of a single lifetime.

Large language models (LLMs) and vision language models (VLMs), such as DeepSeek R1,OpenAI o3, and Gemini 2.5 Pro, have demonstrated remarkable reasoning capabilities across logical inference, problem solving, and decision making. However, spatial reasoning:a fundamental component of human cognition that includes mental rotation, navigation, and spatial relationship comprehension remains a significant challenge for current advanced VLMs. We hypothesize that imagination, the internal simulation of spatial states, is the dominant reasoning mechanism within a spatial world model. To test this hypothesis and systematically probe current VLM spatial reasoning mechanisms, we introduce SpatiaLite, a fully synthetic benchmark that jointly measures spatial reasoning accuracy and reasoning efficiency. Comprehensive experiments reveal three key findings. First, advanced VLMs predominantly rely on linguistic representations for reasoning and imagination, resulting in significant deficiencies on visual centric tasks that demand perceptual spatial relations and 3D geometry transformations such as mental rotation or projection prediction. Second, advanced VLMs exhibit severe inefficiency in their current spatial reasoning mechanisms, with token usage growing rapidly as transformation complexity increases. Third, we propose an Imagery Driven Framework (IDF) for data synthesis and training, which can implicitly construct an internal world model that is critical for spatial reasoning in VLMs. Building on SpatiaLite, this work delineates the spatial reasoning limits and patterns of advanced VLMs, identifies key shortcomings, and informs future advances

Policy Learning (ConPoLe) that explicitly optimizes the InfoNCE loss, which lower bounds the mutual information between the current state and next states that continue on a path to the solution.

Many sequential decision-making problems can be formulated as shortest-path problems, where the objective is to reach a goal state from a given starting state. Heuristic search is a standard approach for solving such problems, relying on a heuristic function to estimate the cost to the goal from any given state. Recent approaches leverage reinforcement learning to learn heuristics by applying deep approximate value iteration. These methods typically rely on single-step Bellman updates, where the heuristic of a state is updated based on its best neighbor and the corresponding edge cost. This work proposes a generalized approach that enhances both state sampling and heuristic updates by performing limited-horizon searches and updating each state's heuristic based on the shortest path to the search frontier, incorporating both edge costs and the heuristic values of frontier states.