Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required error-correcting capability and required types. Infinite series of block codes with rate approaching a given rational $R$ with $0<R<1$ and relative distance over length approaching $(1-R)$ are designed. These can be designed over fields of given characteristic $p$ or over fields of prime order and can be specified to be of a particular type such as (i) dual-containing under Euclidean inner product, (ii) dual-containing under Hermitian inner product, (iii) quantum error-correcting, (iv) linear complementary dual (LCD). Convolutional codes to required rate and distance are designed and infinite series of convolutional codes with rate approaching a given rational $R$ and distance over length approaching $2(1-R)$. Properties, including distances, are shown algebraically and algebraic explicit efficient decoding methods are known.
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Device-to-device (D2D) communications is expected to be a critical enabler of distributed computing in edge networks at scale. A key challenge in providing this capability is the requirement for judicious management of the heterogeneous communication and computation resources that exist at the edge to meet processing needs. In this paper, we develop an optimization methodology that considers the network topology jointly with device and network resource allocation to minimize total D2D overhead, which we quantify in terms of time and energy required for task processing. Variables in our model include task assignment, CPU allocation, subchannel selection, and beamforming design for multiple-input multiple-output (MIMO) wireless devices. We propose two methods to solve the resulting non-convex mixed integer program: semi-exhaustive search optimization, which represents a "best-effort" at obtaining the optimal solution, and efficient alternate optimization, which is more computationally efficient. As a component of these two methods, we develop a novel coordinated beamforming algorithm which we show obtains the optimal beamformer for a common receiver characteristic. Through numerical experiments, we find that our methodology yields substantial improvements in network overhead compared with local computation and partially optimized methods, which validates our joint optimization approach. Further, we find that the efficient alternate optimization scales well with the number of nodes, and thus can be a practical solution for D2D computing in large networks.
Consensus is a common method for computing a function of the data distributed among the nodes of a network. Of particular interest is distributed average consensus, whereby the nodes iteratively compute the sample average of the data stored at all the nodes of the network using only near-neighbor communications. In real-world scenarios, these communications must undergo quantization, which introduces distortion to the internode messages. In this thesis, a model for the evolution of the network state statistics at each iteration is developed under the assumptions of Gaussian data and additive quantization error. It is shown that minimization of the communication load in terms of aggregate source coding rate can be posed as a generalized geometric program, for which an equivalent convex optimization can efficiently solve for the global minimum. Optimization procedures are developed for rate-distortion-optimal vector quantization, uniform entropy-coded scalar quantization, and fixed-rate uniform quantization. Numerical results demonstrate the performance of these approaches. For small numbers of iterations, the fixed-rate optimizations are verified using exhaustive search. Comparison to the prior art suggests competitive performance under certain circumstances but strongly motivates the incorporation of more sophisticated coding strategies, such as differential, predictive, or Wyner-Ziv coding.
Neural networks have shown tremendous growth in recent years to solve numerous problems. Various types of neural networks have been introduced to deal with different types of problems. However, the main goal of any neural network is to transform the non-linearly separable input data into more linearly separable abstract features using a hierarchy of layers. These layers are combinations of linear and nonlinear functions. The most popular and common non-linearity layers are activation functions (AFs), such as Logistic Sigmoid, Tanh, ReLU, ELU, Swish and Mish. In this paper, a comprehensive overview and survey is presented for AFs in neural networks for deep learning. Different classes of AFs such as Logistic Sigmoid and Tanh based, ReLU based, ELU based, and Learning based are covered. Several characteristics of AFs such as output range, monotonicity, and smoothness are also pointed out. A performance comparison is also performed among 18 state-of-the-art AFs with different networks on different types of data. The insights of AFs are presented to benefit the researchers for doing further research and practitioners to select among different choices. The code used for experimental comparison is released at: \url{//github.com/shivram1987/ActivationFunctions}.
Neural Radiance Fields (NeRF) have recently gained a surge of interest within the computer vision community for its power to synthesize photorealistic novel views of real-world scenes. One limitation of NeRF, however, is its requirement of accurate camera poses to learn the scene representations. In this paper, we propose Bundle-Adjusting Neural Radiance Fields (BARF) for training NeRF from imperfect (or even unknown) camera poses -- the joint problem of learning neural 3D representations and registering camera frames. We establish a theoretical connection to classical image alignment and show that coarse-to-fine registration is also applicable to NeRF. Furthermore, we show that na\"ively applying positional encoding in NeRF has a negative impact on registration with a synthesis-based objective. Experiments on synthetic and real-world data show that BARF can effectively optimize the neural scene representations and resolve large camera pose misalignment at the same time. This enables view synthesis and localization of video sequences from unknown camera poses, opening up new avenues for visual localization systems (e.g. SLAM) and potential applications for dense 3D mapping and reconstruction.
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.
Lots of learning tasks require dealing with graph data which contains rich relation information among elements. Modeling physics system, learning molecular fingerprints, predicting protein interface, and classifying diseases require that a model to learn from graph inputs. In other domains such as learning from non-structural data like texts and images, reasoning on extracted structures, like the dependency tree of sentences and the scene graph of images, is an important research topic which also needs graph reasoning models. Graph neural networks (GNNs) are connectionist models that capture the dependence of graphs via message passing between the nodes of graphs. Unlike standard neural networks, graph neural networks retain a state that can represent information from its neighborhood with an arbitrary depth. Although the primitive graph neural networks have been found difficult to train for a fixed point, recent advances in network architectures, optimization techniques, and parallel computation have enabled successful learning with them. In recent years, systems based on graph convolutional network (GCN) and gated graph neural network (GGNN) have demonstrated ground-breaking performance on many tasks mentioned above. In this survey, we provide a detailed review over existing graph neural network models, systematically categorize the applications, and propose four open problems for future research.
We present an end-to-end framework for solving the Vehicle Routing Problem (VRP) using reinforcement learning. In this approach, we train a single model that finds near-optimal solutions for problem instances sampled from a given distribution, only by observing the reward signals and following feasibility rules. Our model represents a parameterized stochastic policy, and by applying a policy gradient algorithm to optimize its parameters, the trained model produces the solution as a sequence of consecutive actions in real time, without the need to re-train for every new problem instance. On capacitated VRP, our approach outperforms classical heuristics and Google's OR-Tools on medium-sized instances in solution quality with comparable computation time (after training). We demonstrate how our approach can handle problems with split delivery and explore the effect of such deliveries on the solution quality. Our proposed framework can be applied to other variants of the VRP such as the stochastic VRP, and has the potential to be applied more generally to combinatorial optimization problems.
In recent years, many publications showed that convolutional neural network based features can have a superior performance to engineered features. However, not much effort was taken so far to extract local features efficiently for a whole image. In this paper, we present an approach to compute patch-based local feature descriptors efficiently in presence of pooling and striding layers for whole images at once. Our approach is generic and can be applied to nearly all existing network architectures. This includes networks for all local feature extraction tasks like camera calibration, Patchmatching, optical flow estimation and stereo matching. In addition, our approach can be applied to other patch-based approaches like sliding window object detection and recognition. We complete our paper with a speed benchmark of popular CNN based feature extraction approaches applied on a whole image, with and without our speedup, and example code (for Torch) that shows how an arbitrary CNN architecture can be easily converted by our approach.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
Querying graph structured data is a fundamental operation that enables important applications including knowledge graph search, social network analysis, and cyber-network security. However, the growing size of real-world data graphs poses severe challenges for graph databases to meet the response-time requirements of the applications. Planning the computational steps of query processing - Query Planning - is central to address these challenges. In this paper, we study the problem of learning to speedup query planning in graph databases towards the goal of improving the computational-efficiency of query processing via training queries.We present a Learning to Plan (L2P) framework that is applicable to a large class of query reasoners that follow the Threshold Algorithm (TA) approach. First, we define a generic search space over candidate query plans, and identify target search trajectories (query plans) corresponding to the training queries by performing an expensive search. Subsequently, we learn greedy search control knowledge to imitate the search behavior of the target query plans. We provide a concrete instantiation of our L2P framework for STAR, a state-of-the-art graph query reasoner. Our experiments on benchmark knowledge graphs including DBpedia, YAGO, and Freebase show that using the query plans generated by the learned search control knowledge, we can significantly improve the speed of STAR with negligible loss in accuracy.
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