While humans show remarkable proficiency in solving visual puzzles, machines often fall short due to the complex combinatorial nature of such tasks. Consequently, there is a growing interest in developing computational methods for the automatic solution of different puzzles, especially through deep learning approaches. The Tangram, an ancient Chinese puzzle, challenges players to arrange seven polygonal pieces to construct different patterns. Despite its apparent simplicity, solving the Tangram is considered an NP-complete problem, being a challenge even for the most sophisticated algorithms. Moreover, ensuring the generality and adaptability of machine learning models across different Tangram arrangements and complexities is an ongoing research problem. In this paper, we introduce a generative model specifically designed to solve the Tangram. Our model competes favorably with previous methods regarding accuracy while delivering fast inferences. It incorporates a novel loss function that integrates pixel-based information with geometric features, promoting a deeper understanding of the spatial relationships between pieces. Unlike previous approaches, our model takes advantage of the geometric properties of the Tangram to formulate a solving strategy, exploiting its inherent properties only through exposure to training data rather than through direct instruction. Extending the proposed loss function, we present a novel evaluation metric as a better fitting measure for assessing Tangram solutions than previous metrics. We further provide a new dataset containing more samples than others reported in the literature. Our findings highlight the potential of deep learning approaches in geometric problem domains.

Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear operator; an important special case is random design regression. We derive sharp rates of estimation for arbitrary compact elliptical parameter sets and demonstrate how they depend on the distribution of the random linear operator. Our main result is a functional that characterizes the minimax rate of estimation in terms of the noise level, the law of the random operator, and elliptical norms that define the error metric and the parameter space. This nonasymptotic result is sharp up to an explicit universal constant, and it becomes asymptotically exact as the radius of the parameter space is allowed to grow. We demonstrate the generality of the result by applying it to both parametric and nonparametric regression problems, including those involving distribution shift or dependent covariates.

The performance of domain adaptation technologies has not yet reached an ideal level in the current 3D object detection field for autonomous driving, which is mainly due to significant differences in the size of vehicles, as well as the environments they operate in when applied across domains. These factors together hinder the effective transfer and application of knowledge learned from specific datasets. Since the existing evaluation metrics are initially designed for evaluation on a single domain by calculating the 2D or 3D overlap between the prediction and ground-truth bounding boxes, they often suffer from the overfitting problem caused by the size differences among datasets. This raises a fundamental question related to the evaluation of the 3D object detection models' cross-domain performance: Do we really need models to maintain excellent performance in their original 3D bounding boxes after being applied across domains? From a practical application perspective, one of our main focuses is actually on preventing collisions between vehicles and other obstacles, especially in cross-domain scenarios where correctly predicting the size of vehicles is much more difficult. In other words, as long as a model can accurately identify the closest surfaces to the ego vehicle, it is sufficient to effectively avoid obstacles. In this paper, we propose two metrics to measure 3D object detection models' ability of detecting the closer surfaces to the sensor on the ego vehicle, which can be used to evaluate their cross-domain performance more comprehensively and reasonably. Furthermore, we propose a refinement head, named EdgeHead, to guide models to focus more on the learnable closer surfaces, which can greatly improve the cross-domain performance of existing models not only under our new metrics, but even also under the original BEV/3D metrics. Our code is available at https://github.com/Galaxy-ZRX/EdgeHead.

Factorization machine (FM) variants are widely used in recommendation systems that operate under strict throughput and latency requirements, such as online advertising systems. FMs have two prominent strengths. First, is their ability to model pairwise feature interactions while being resilient to data sparsity by learning factorized representations. Second, their computational graphs facilitate fast inference and training. Moreover, when items are ranked as a part of a query for each incoming user, these graphs facilitate computing the portion stemming from the user and context fields only once per query. Thus, the computational cost for each ranked item is proportional only to the number of fields that vary among the ranked items. Consequently, in terms of inference cost, the number of user or context fields is practically unlimited. More advanced variants of FMs, such as field-aware and fieldweighted FMs, provide better accuracy by learning a representation of field-wise interactions, but require computing all pairwise interaction terms explicitly. In particular, the computational cost during inference is proportional to the square of the number of fields, including user, context, and item. When the number of fields is large, this is prohibitive in systems with strict latency constraints, and imposes a limit on the number of user and context fields for a given computational budget. To mitigate this caveat, heuristic pruning of low intensity field interactions is commonly used to accelerate inference. In this work we propose an alternative to the pruning heuristic in field-weighted FMs using a diagonal plus symmetric lowrank decomposition. Our technique reduces the computational cost of inference, by allowing it to be proportional to the number of item fields only. Using a set of experiments on real-world datasets, we show that aggressive rank reduction outperforms similarly aggressive pruning, both in terms of accuracy and item recommendation speed. Beyond computational complexity analysis, we corroborate our claim of faster inference experimentally, both via a synthetic test, and by having deployed our solution