Recent work showed that large diffusion models can be reused as highly precise monocular depth estimators by casting depth estimation as an image-conditional image generation task. While the proposed model achieved state-of-the-art results, high computational demands due to multi-step inference limited its use in many scenarios. In this paper, we show that the perceived inefficiency was caused by a flaw in the inference pipeline that has so far gone unnoticed. The fixed model performs comparably to the best previously reported configuration while being more than 200x faster. To optimize for downstream task performance, we perform end-to-end fine-tuning on top of the single-step model with task-specific losses and get a deterministic model that outperforms all other diffusion-based depth and normal estimation models on common zero-shot benchmarks. We surprisingly find that this fine-tuning protocol also works directly on Stable Diffusion and achieves comparable performance to current state-of-the-art diffusion-based depth and normal estimation models, calling into question some of the conclusions drawn from prior works.

With this paper we present an extension of our recent ISSAC paper about computations of Groebner(-Shirshov) bases over free associative algebras Z . We present all the needed proofs in details, add a part on the direct treatment of the ring Z/mZ as well as new examples and applications to e.g. Iwahori-Hecke algebras.The extension of Groebner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance.Groebner and Groebner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate ourselves on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behavior of Groebner bases between free algebras and algebras, close to commutative.Even the process of the formation of critical pairs has to be reengineered, together with the implementing the criteria for their quick discarding.We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Groebner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation.

The extension of Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate ourselves on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behavior of Gröbner bases between free algebras and algebras, close to commutative. Even the formation of critical pairs has to be reengineered, together with the criteria for their quick discarding. We present an implementation of algorithms in the Singular subsystem called Letterplace, which internally uses Letterplace techniques (and Letterplace Gröbner bases), due to La Scala and Levandovskyy. Interesting examples accompany our presentation.

We present the newest release of the subsystem of Singular called Letterplace which exists since 2009. It is devoted to computations with finitely presented associative algebras over fields and offers Gröbner(-Shirshov) bases over free algebras via the Letterplace correspondence of La Scala and Levandovskyy. This allows to use highly tuned commutative data structures internally and to reuse parts of existing algorithms in the non-commutative situation. The present version has been deeply reengineered, based on the experience with earlier and experimental versions. We offer an unprecedented functionality, some of which for the first time in the history of computer algebra. In particular, we present tools for elimination theory (via truncated Gröbner bases and via supporting several kinds of elimination orderings), dimension theory (Gel'fand-Kirillov and global dimension), and for homological algebra (such as syzygy bimodules and lifts for ideals and bimodules) to name a few. Another article in this issue is devoted to the extension of Gröbner bases to the coefficients in principal ideal rings including Z, which is also a part of this release. We report on comparison with other systems and on some advances in the theory. Quite nontrivial examples illustrate the abilities of the system.