Hierarchical Clustering has been studied and used extensively as a method for analysis of data. More recently, Dasgupta [2016] defined a precise objective function. Given a set of $n$ data points with a weight function $w_{i,j}$ for each two items $i$ and $j$ denoting their similarity/dis-similarity, the goal is to build a recursive (tree like) partitioning of the data points (items) into successively smaller clusters. He defined a cost function for a tree $T$ to be $Cost(T) = \sum_{i,j \in [n]} \big(w_{i,j} \times |T_{i,j}| \big)$ where $T_{i,j}$ is the subtree rooted at the least common ancestor of $i$ and $j$ and presented the first approximation algorithm for such clustering. Then Moseley and Wang [2017] considered the dual of Dasgupta's objective function for similarity-based weights and showed that both random partitioning and average linkage have approximation ratio $1/3$ which has been improved in a series of works to $0.585$ [Alon et al. 2020]. Later Cohen-Addad et al. [2019] considered the same objective function as Dasgupta's but for dissimilarity-based metrics, called $Rev(T)$. It is shown that both random partitioning and average linkage have ratio $2/3$ which has been only slightly improved to $0.667078$ [Charikar et al. SODA2020]. Our first main result is to consider $Rev(T)$ and present a more delicate algorithm and careful analysis that achieves approximation $0.71604$. We also introduce a new objective function for dissimilarity-based clustering. For any tree $T$, let $H_{i,j}$ be the number of $i$ and $j$'s common ancestors. Intuitively, items that are similar are expected to remain within the same cluster as deep as possible. So, for dissimilarity-based metrics, we suggest the cost of each tree $T$, which we want to minimize, to be $Cost_H(T) = \sum_{i,j \in [n]} \big(w_{i,j} \times H_{i,j} \big)$. We present a $1.3977$-approximation for this objective.
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我(wo)們給定x，函(han)數都會輸出(chu)一個(ge)f(X)，這個(ge)輸出(chu)的f(X)與真(zhen)實值Y可能(neng)是相同的，也可能(neng)是不同的，為了表示擬合的好壞，就(jiu)用一個(ge)函(han)數來度量擬合的程度。這個(ge)函(han)數就(jiu)稱(cheng)為損失函(han)數(loss function)，或者叫代價函(han)數(cost function)
Hamilton and Moitra (2021) showed that, in certain regimes, it is not possible to accelerate Riemannian gradient descent in the hyperbolic plane if we restrict ourselves to algorithms which make queries in a (large) bounded domain and which receive gradients and function values corrupted by a (small) amount of noise. We show that acceleration remains unachievable for any deterministic algorithm which receives exact gradient and function-value information (unbounded queries, no noise). Our results hold for the classes of strongly and nonstrongly geodesically convex functions, and for a large class of Hadamard manifolds including hyperbolic spaces and the symmetric space $\mathrm{SL}(n) / \mathrm{SO}(n)$ of positive definite $n \times n$ matrices of determinant one. This cements a surprising gap between the complexity of convex optimization and geodesically convex optimization: for hyperbolic spaces, Riemannian gradient descent is optimal on the class of smooth and and strongly geodesically convex functions, in the regime where the condition number scales with the radius of the optimization domain. The key idea for proving the lower bound consists of perturbing the hard functions of Hamilton and Moitra (2021) with sums of bump functions chosen by a resisting oracle.
The analysis of data streams has received considerable attention over the past few decades due to sensors, social media, etc. It aims to recognize patterns in an unordered, infinite, and evolving stream of observations. Clustering this type of data requires some restrictions in time and memory. This paper introduces a new data stream clustering method (IMOC-Stream). This method, unlike the other clustering algorithms, uses two different objective functions to capture different aspects of the data. The goal of IMOC-Stream is to: 1) reduce computation time by using idle times to apply genetic operations and enhance the solution. 2) reduce memory allocation by introducing a new tree synopsis. 3) find arbitrarily shaped clusters by using a multi-objective framework. We conducted an experimental study with high dimensional stream datasets and compared them to well-known stream clustering techniques. The experiments show the ability of our method to partition the data stream in arbitrarily shaped, compact, and well-separated clusters while optimizing the time and memory. Our method also outperformed most of the stream algorithms in terms of NMI and ARAND measures.
In addition to being extremely non-linear, modern problems require millions if not billions of parameters to solve or at least to get a good approximation of the solution, and neural networks are known to assimilate that complexity by deepening and widening their topology in order to increase the level of non-linearity needed for a better approximation. However, compact topologies are always preferred to deeper ones as they offer the advantage of using less computational units and less parameters. This compacity comes at the price of reduced non-linearity and thus, of limited solution search space. We propose the 1-Dimensional Polynomial Neural Network (1DPNN) model that uses automatic polynomial kernel estimation for 1-Dimensional Convolutional Neural Networks (1DCNNs) and that introduces a high degree of non-linearity from the first layer which can compensate the need for deep and/or wide topologies. We show that this non-linearity enables the model to yield better results with less computational and spatial complexity than a regular 1DCNN on various classification and regression problems related to audio signals, even though it introduces more computational and spatial complexity on a neuronal level. The experiments were conducted on three publicly available datasets and demonstrate that, on the problems that were tackled, the proposed model can extract more relevant information from the data than a 1DCNN in less time and with less memory.
In this paper, we propose a one-stage online clustering method called Contrastive Clustering (CC) which explicitly performs the instance- and cluster-level contrastive learning. To be specific, for a given dataset, the positive and negative instance pairs are constructed through data augmentations and then projected into a feature space. Therein, the instance- and cluster-level contrastive learning are respectively conducted in the row and column space by maximizing the similarities of positive pairs while minimizing those of negative ones. Our key observation is that the rows of the feature matrix could be regarded as soft labels of instances, and accordingly the columns could be further regarded as cluster representations. By simultaneously optimizing the instance- and cluster-level contrastive loss, the model jointly learns representations and cluster assignments in an end-to-end manner. Extensive experimental results show that CC remarkably outperforms 17 competitive clustering methods on six challenging image benchmarks. In particular, CC achieves an NMI of 0.705 (0.431) on the CIFAR-10 (CIFAR-100) dataset, which is an up to 19\% (39\%) performance improvement compared with the best baseline.
Spectral clustering (SC) is a popular clustering technique to find strongly connected communities on a graph. SC can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is expensive and, since clustering results are graph-specific, pooling methods based on SC must perform a new optimization for each new sample. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem and train a GNN to compute cluster assignments that minimize this objective. Our GNN-based implementation is differentiable, does not require to compute the spectral decomposition, and learns a clustering function that can be quickly evaluated on out-of-sample graphs. From the proposed clustering method, we design a graph pooling operator that overcomes some important limitations of state-of-the-art graph pooling techniques and achieves the best performance in several supervised and unsupervised tasks.
Attributed graph clustering is challenging as it requires joint modelling of graph structures and node attributes. Recent progress on graph convolutional networks has proved that graph convolution is effective in combining structural and content information, and several recent methods based on it have achieved promising clustering performance on some real attributed networks. However, there is limited understanding of how graph convolution affects clustering performance and how to properly use it to optimize performance for different graphs. Existing methods essentially use graph convolution of a fixed and low order that only takes into account neighbours within a few hops of each node, which underutilizes node relations and ignores the diversity of graphs. In this paper, we propose an adaptive graph convolution method for attributed graph clustering that exploits high-order graph convolution to capture global cluster structure and adaptively selects the appropriate order for different graphs. We establish the validity of our method by theoretical analysis and extensive experiments on benchmark datasets. Empirical results show that our method compares favourably with state-of-the-art methods.
Clustering is an essential data mining tool that aims to discover inherent cluster structure in data. For most applications, applying clustering is only appropriate when cluster structure is present. As such, the study of clusterability, which evaluates whether data possesses such structure, is an integral part of cluster analysis. However, methods for evaluating clusterability vary radically, making it challenging to select a suitable measure. In this paper, we perform an extensive comparison of measures of clusterability and provide guidelines that clustering users can reference to select suitable measures for their applications.
In this paper, we propose an improved quantitative evaluation framework for Generative Adversarial Networks (GANs) on generating domain-specific images, where we improve conventional evaluation methods on two levels: the feature representation and the evaluation metric. Unlike most existing evaluation frameworks which transfer the representation of ImageNet inception model to map images onto the feature space, our framework uses a specialized encoder to acquire fine-grained domain-specific representation. Moreover, for datasets with multiple classes, we propose Class-Aware Frechet Distance (CAFD), which employs a Gaussian mixture model on the feature space to better fit the multi-manifold feature distribution. Experiments and analysis on both the feature level and the image level were conducted to demonstrate improvements of our proposed framework over the recently proposed state-of-the-art FID method. To our best knowledge, we are the first to provide counter examples where FID gives inconsistent results with human judgments. It is shown in the experiments that our framework is able to overcome the shortness of FID and improves robustness. Code will be made available.
Image segmentation is the process of partitioning the image into significant regions easier to analyze. Nowadays, segmentation has become a necessity in many practical medical imaging methods as locating tumors and diseases. Hidden Markov Random Field model is one of several techniques used in image segmentation. It provides an elegant way to model the segmentation process. This modeling leads to the minimization of an objective function. Conjugate Gradient algorithm (CG) is one of the best known optimization techniques. This paper proposes the use of the Conjugate Gradient algorithm (CG) for image segmentation, based on the Hidden Markov Random Field. Since derivatives are not available for this expression, finite differences are used in the CG algorithm to approximate the first derivative. The approach is evaluated using a number of publicly available images, where ground truth is known. The Dice Coefficient is used as an objective criterion to measure the quality of segmentation. The results show that the proposed CG approach compares favorably with other variants of Hidden Markov Random Field segmentation algorithms.
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