A localisation theorem for singularity categories of proper dg algebras

Given a recollement of three proper dg algebras over a noetherian commutative ring, e.g. three algebras which are finitely generated over the base ring, which extends one step downwards, it is shown that there is a short exact sequence of their singularity categories. This allows us to recover and generalise some known results on singularity categories of finite-dimensional algebras.Comment: Section 3 is new, and in Section 4 the base ring is changed from a field to a commutative noetherian ring and in Section 6 the base ring is changed from a field to an arbitrary commutative rin

Solving Inverse Obstacle Scattering Problem with Latent Surface Representations

We propose a novel iterative numerical method to solve the three-dimensional inverse obstacle scattering problem of recovering the shape of the obstacle from far-field measurements. To address the inherent ill-posed nature of the inverse problem, we advocate the use of a trained latent representation of surfaces as the generative prior. This prior enjoys excellent expressivity within the given class of shapes, and meanwhile, the latent dimensionality is low, which greatly facilitates the computation. Thus, the admissible manifold of surfaces is realistic and the resulting optimization problem is less ill-posed. We employ the shape derivative to evolve the latent surface representation, by minimizing the loss, and we provide a local convergence analysis of a gradient descent type algorithm to a stationary point of the loss. We present several numerical examples, including also backscattered and phaseless data, to showcase the effectiveness of the proposed algorithm

Learning Robust Representation for Joint Grading of Ophthalmic Diseases via Adaptive Curriculum and Feature Disentanglement

Diabetic retinopathy (DR) and diabetic macular edema (DME) are leading causes of permanent blindness worldwide. Designing an automatic grading system with good generalization ability for DR and DME is vital in clinical practice. However, prior works either grade DR or DME independently, without considering internal correlations between them, or grade them jointly by shared feature representation, yet ignoring potential generalization issues caused by difficult samples and data bias. Aiming to address these problems, we propose a framework for joint grading with the dynamic difficulty-aware weighted loss (DAW) and the dual-stream disentangled learning architecture (DETACH). Inspired by curriculum learning, DAW learns from simple samples to difficult samples dynamically via measuring difficulty adaptively. DETACH separates features of grading tasks to avoid potential emphasis on the bias. With the addition of DAW and DETACH, the model learns robust disentangled feature representations to explore internal correlations between DR and DME and achieve better grading performance. Experiments on three benchmarks show the effectiveness and robustness of our framework under both the intra-dataset and cross-dataset tests.Comment: Accepted by MICCAI2

Recollements of partially wrapped Fukaya categories and surface cuts

In this paper we use recollements to investigate partially wrapped Fukaya categories of surfaces with marked points. In particular, we show that cutting surfaces gives rise to recollements of the corresponding partially wrapped Fukaya categories. Our approach is based on the fact that the partially wrapped Fukaya category of a surface with marked points is triangle equivalent to the perfect derived category of a homologically smooth and proper graded gentle algebra with zero differential as shown by Haiden, Katzarkov and Kontsevich. Using this, we study particular generators of partially wrapped Fukaya categories, namely full exceptional sequences, silting objects and simple-minded collections. In particular, we fully characterise the existence of full exceptional sequences and we give an example of a partially wrapped Fukaya category which does not admit a silting object, that is a generator with no positive self-extensions.Comment: 30 pages. Comments are welcome v2: changes to title, abstract and introductio