More precisely, we use batches of size 2. Each batch contains one patch with the foreground oversampled. Furthermore, we split each silo's data into training and validation data with 80% and 20% split, respectively. All this pre-processing and patching is done using the nnU-Net library [IJK+21]. Loss function We use the same loss function as proposed by nnU-Net [IJK+21] for the KiTS19 dataset which is based on DICE [Dic45] and on the Cross Entropy loss.

In this section, we review some key results on the Wasserstein distance. Wpp Rπ(t,θi),Rρ(t,θi), (4) where the approximation comes from using Monte-Carlo integration by samplingθi uniformly in SD 1 [2]. M,M is the number of points used to approximate the integral. Calculating the Wasserstein distance with the empirical distribution function is computationally attractive. To do that, we first sortxms in an ascending order, such thatxi[m] xi[m+1], where i[m]istheindexofthesortedxms. Hamiltonian Monte Carlo (HMC)[24]isahighly-efficient MarkovChain Monte Carlo (MCMC) method used to generate samples from the posteriorw p(w|y).

Different inner loss and outer loss other than what we used can be instantiated in the subclass.

Baseline Model As a baseline, we use the aforementioned linear Cox proportional hazard model[Cox72].

However, since this proof relies on the existence of a convergent subsequence, their proof does not reveal any rate forglobal convergence.

Motion blur caused by camera shake, particularly under large or rotational movements, remains a major challenge in image restoration. We propose a deep learning framework that jointly estimates the latent sharp image and the underlying camera motion trajectory from a single blurry image. Our method leverages the Projective Motion Blur Model (PMBM), implemented efficiently using a differentiable blur creation module compatible with modern networks. A neural network predicts a full 3D rotation trajectory, which guides a model-based restoration network trained end-to-end. This modular architecture provides interpretability by revealing the camera motion that produced the blur. Moreover, this trajectory enables the reconstruction of the sequence of sharp images that generated the observed blurry image. To further refine results, we optimize the trajectory post-inference via a reblur loss, improving consistency between the blurry input and the restored output. Extensive experiments show that our method achieves state-of-the-art performance on both synthetic and real datasets, particularly in cases with severe or spatially variant blur, where end-to-end deblurring networks struggle. Code and trained models are available at https://github.com/GuillermoCarbajal/Blur2Seq/

Consider the following system with a generic R.H.S-X We also added the approach of Blackbox [25], which also deals with a combinatorial optimization problem with a linear objective.

Different inner loss and outer loss other than what we used can be instantiated in the subclass.

Program synthesis with Large Language Models (LLMs) suffers from a "near-miss syndrome": the generated code closely resembles a correct solution but fails unit tests due to minor errors. We address this with a multi-agent framework called Synthesize, Execute, Instruct, Debug, and Repair (SEIDR). Effectively applying SEIDR to instruction-tuned LLMs requires determining (a) optimal prompts for LLMs, (b) what ranking algorithm selects the best programs in debugging rounds, and (c) balancing the repair of unsuccessful programs with the generation of new ones. We empirically explore these trade-offs by comparing replace-focused, repair-focused, and hybrid debug strategies. We also evaluate lexicase and tournament selection to rank candidates in each generation. On Program Synthesis Benchmark 2 (PSB2), our framework outperforms both conventional use of OpenAI Codex without a repair phase and traditional genetic programming approaches. SEIDR outperforms the use of an LLM alone, solving 18 problems in C++ and 20 in Python on PSB2 at least once across experiments. To assess generalizability, we employ GPT-3.5 and Llama 3 on the PSB2 and HumanEval-X benchmarks. Although SEIDR with these models does not surpass current state-of-the-art methods on the Python benchmarks, the results on HumanEval-C++ are promising. SEIDR with Llama 3-8B achieves an average pass@100 of 84.2%. Across all SEIDR runs, 163 of 164 problems are solved at least once with GPT-3.5 in HumanEval-C++, and 162 of 164 with the smaller Llama 3-8B. We conclude that SEIDR effectively overcomes the near-miss syndrome in program synthesis with LLMs.