Tensor compilers, essential for generating efficient code for deep learning models across various applications, employ tensor graph rewrites as one of the key optimizations. These rewrites optimize tensor computational graphs with the expectation of preserving semantics for tensors of arbitrary rank and size. Despite this expectation, to the best of our knowledge, there does not exist a fully automated verification system to prove the soundness of these rewrites for tensors of arbitrary rank and size. Previous works, while successful in verifying rewrites with tensors of concrete rank, do not provide guarantees in the unbounded setting. To fill this gap, we introduce T ensor R ight , the first automatic verification system that can verify tensor graph rewrites for input tensors of arbitrary rank and size. We introduce a core language, T ensor R ight DSL, to represent rewrite rules using a novel axis definition, called aggregated-axis , which allows us to reason about an unbounded number of axes. We achieve unbounded verification by proving that there exists a bound on tensor ranks, under which bounded verification of all instances implies the correctness of the rewrite rule in the unbounded setting. We derive an algorithm to compute this rank using the denotational semantics of T ensor R ight DSL. T ensor R ight employs this algorithm to generate a finite number of bounded-verification proof obligations, which are then dispatched to an SMT solver using symbolic execution to automatically verify the correctness of the rewrite rules. We evaluate T ensor R ight ’s verification capabilities by implementing rewrite rules present in XLA ’s algebraic simplifier. The results demonstrate that T ensor R ight can prove the correctness of 115 out of 175 rules in their full generality, while the closest automatic, bounded -verification system can express only 18 of these rules.

Sagittal spino-pelvic balance has been increasingly emphasized in hip surgery. The conversion between standing and sitting, characterized by complementary pelvic angles (pelvic tilt, pt and sacral slope, ss), involves a congruent sagittal spino-pelvic relationship. Hence, the changes of complementary pelvic angles pt, ss between standing and sitting could reflect the mechanism of sagittal spino-pelvic balance, and should be analyzed in evidence-based hip surgery planning. To this end, we propose a novel cross LSTM (C-LSTM) framework embedding the conversion between standing and sitting by cross-mapping, to predict the changes of complementary pelvic pt, ss between standing and sitting. Furthermore, to introduce the prior knowledge of the invariance of pelvic incidence, pi, two dual C-LSTMs are integrated to construct a much more powerful Fused C-LSTM. We have conducted extensive experiments on the sagittal standing-sitting dataset for the comprehensive evaluation of the proposed framework. Even in a small samples, Fused C-LSTM can achieve low prediction errors and high correlation between predicted and actual values. Notably, just based on static standing or sitting X-ray, Fused C-LSTM can obtain the change of complementary pt, ss between standing and sitting to assist in formulating a surgical hip plan that conforms to the sagittal spino-pelvic balance.