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We present Bayesian Diffusion Models (BDM), a prediction algorithm that performs effective Bayesian inference by tightly coupling the top-down (prior) information with the bottom-up (data-driven) procedure via joint diffusion processes. We show the effectiveness of BDM on the 3D shape reconstruction task. Compared to prototypical deep learning data-driven approaches trained on paired (supervised) data-labels (e.g. image-point clouds) datasets, our BDM brings in rich prior information from standalone labels (e.g. point clouds) to improve the bottom-up 3D reconstruction. As opposed to the standard Bayesian frameworks where explicit prior and likelihood are required for the inference, BDM performs seamless information fusion via coupled diffusion processes with learned gradient computation networks. The specialty of our BDM lies in its capability to engage the active and effective information exchange and fusion of the top-down and bottom-up processes where each itself is a diffusion process. We demonstrate state-of-the-art results on both synthetic and real-world benchmarks for 3D shape reconstruction.

An individualized treatment rule (ITR) is a decision rule that recommends treatments for patients based on their individual feature variables. In many practices, the ideal ITR for the primary outcome is also expected to cause minimal harm to other secondary outcomes. Therefore, our objective is to learn an ITR that not only maximizes the value function for the primary outcome, but also approximates the optimal rule for the secondary outcomes as closely as possible. To achieve this goal, we introduce a fusion penalty to encourage the ITRs based on different outcomes to yield similar recommendations. Two algorithms are proposed to estimate the ITR using surrogate loss functions. We prove that the agreement rate between the estimated ITR of the primary outcome and the optimal ITRs of the secondary outcomes converges to the true agreement rate faster than if the secondary outcomes are not taken into consideration. Furthermore, we derive the non-asymptotic properties of the value function and misclassification rate for the proposed method. Finally, simulation studies and a real data example are used to demonstrate the finite-sample performance of the proposed method.

We consider the planar dynamic convex hull problem. In the literature, solutions exist supporting the insertion and deletion of points in poly-logarithmic time and various queries on the convex hull of the current set of points in logarithmic time. If arbitrary insertion and deletion of points are allowed, constant time updates and fast queries are known to be impossible. This paper considers two restricted cases where worst-case constant time updates and logarithmic time queries are possible. We assume all updates are performed on a deque (double-ended queue) of points. The first case considers the monotonic path case, where all points are sorted in a given direction, say horizontally left-to-right, and only the leftmost and rightmost points can be inserted and deleted. The second case assumes that the points in the deque constitute a simple path. Note that the monotone case is a special case of the simple path case. For both cases, we present solutions supporting deque insertions and deletions in worst-case constant time and standard queries on the convex hull of the points in $O(\log n)$ time, where $n$ is the number of points in the current point set. The convex hull of the current point set can be reported in $O(h+\log n)$ time, where $h$ is the number of edges of the convex hull. For the 1-sided monotone path case, where updates are only allowed on one side, the reporting time can be reduced to $O(h)$, and queries on the convex hull are supported in $O(\log h)$ time. All our time bounds are worst case. In addition, we prove lower bounds that match these time bounds, and thus our results are optimal. For a quick comparison, the previous best update bounds for the simple path problem were amortized $O(\log n)$ time by Friedman, Hershberger, and Snoeyink [SoCG 1989].

In this work, we investigate the effect of momentum on the optimisation trajectory of gradient descent. We leverage a continuous-time approach in the analysis of momentum gradient descent with step size $\gamma$ and momentum parameter $\beta$ that allows us to identify an intrinsic quantity $\lambda = \frac{ \gamma }{ (1 - \beta)^2 }$ which uniquely defines the optimisation path and provides a simple acceleration rule. When training a $2$-layer diagonal linear network in an overparametrised regression setting, we characterise the recovered solution through an implicit regularisation problem. We then prove that small values of $\lambda$ help to recover sparse solutions. Finally, we give similar but weaker results for stochastic momentum gradient descent. We provide numerical experiments which support our claims.

Inspired by the exceptional general intelligence of Large Language Models (LLMs), researchers have begun to explore their application in pioneering the next generation of recommender systems - systems that are conversational, explainable, and controllable. However, existing literature primarily concentrates on integrating domain-specific knowledge into LLMs to enhance accuracy, often neglecting the ability to follow instructions. To address this gap, we initially introduce a collection of supervised learning tasks, augmented with labels derived from a conventional recommender model, aimed at explicitly improving LLMs' proficiency in adhering to recommendation-specific instructions. Subsequently, we develop a reinforcement learning-based alignment procedure to further strengthen LLMs' aptitude in responding to users' intentions and mitigating formatting errors. Through extensive experiments on two real-world datasets, our method markedly advances the capability of LLMs to comply with instructions within recommender systems, while sustaining a high level of accuracy performance.

We study the identification of causal effects, motivated by two improvements to identifiability which can be attained if one knows that some variables in a causal graph are functionally determined by their parents (without needing to know the specific functions). First, an unidentifiable causal effect may become identifiable when certain variables are functional. Second, certain functional variables can be excluded from being observed without affecting the identifiability of a causal effect, which may significantly reduce the number of needed variables in observational data. Our results are largely based on an elimination procedure which removes functional variables from a causal graph while preserving key properties in the resulting causal graph, including the identifiability of causal effects.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.

We consider the problem of explaining the predictions of graph neural networks (GNNs), which otherwise are considered as black boxes. Existing methods invariably focus on explaining the importance of graph nodes or edges but ignore the substructures of graphs, which are more intuitive and human-intelligible. In this work, we propose a novel method, known as SubgraphX, to explain GNNs by identifying important subgraphs. Given a trained GNN model and an input graph, our SubgraphX explains its predictions by efficiently exploring different subgraphs with Monte Carlo tree search. To make the tree search more effective, we propose to use Shapley values as a measure of subgraph importance, which can also capture the interactions among different subgraphs. To expedite computations, we propose efficient approximation schemes to compute Shapley values for graph data. Our work represents the first attempt to explain GNNs via identifying subgraphs explicitly and directly. Experimental results show that our SubgraphX achieves significantly improved explanations, while keeping computations at a reasonable level.

Recently, a considerable literature has grown up around the theme of Graph Convolutional Network (GCN). How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly propagating and updating the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the Knowledge Embedding based Graph Convolutional Network (KE-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge embedding (a.k.a. knowledge graph embedding) methods, and goes beyond. Our theoretical analysis shows that KE-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of KE-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.

Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.