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Due to privacy and security concerns, recent advancements in group fairness advocate for model training regardless of demographic information. However, most methods still require prior knowledge of demographics. In this study, we explore the potential for achieving fairness without compromising its utility when no prior demographics are provided to the training set, namely \emph{harmless Rawlsian fairness}. We ascertain that such a fairness requirement with no prior demographic information essential promotes training losses to exhibit a Dirac delta distribution. To this end, we propose a simple but effective method named VFair to minimize the variance of training losses inside the optimal set of empirical losses. This problem is then optimized by a tailored dynamic update approach that operates in both loss and gradient dimensions, directing the model towards relatively fairer solutions while preserving its intact utility. Our experimental findings indicate that regression tasks, which are relatively unexplored from literature, can achieve significant fairness improvement through VFair regardless of any prior, whereas classification tasks usually do not because of their quantized utility measurements. The implementation of our method is publicly available at \url{https://github.com/wxqpxw/VFair}., Comment: Accepted as a Poster in Neurips 2024

Diffusion models have demonstrated great potential in generating high-quality content for images, natural language, protein domains, etc. However, how to perform user-preferred targeted generation via diffusion models with only black-box target scores of users remains challenging. To address this issue, we first formulate the fine-tuning of the targeted reserve-time stochastic differential equation (SDE) associated with a pre-trained diffusion model as a sequential black-box optimization problem. Furthermore, we propose a novel covariance-adaptive sequential optimization algorithm to optimize cumulative black-box scores under unknown transition dynamics. Theoretically, we prove a $O(\frac{d^2}{\sqrt{T}})$ convergence rate for cumulative convex functions without smooth and strongly convex assumptions. Empirically, experiments on both numerical test problems and target-guided 3D-molecule generation tasks show the superior performance of our method in achieving better target scores.

We propose a novel graph cross network (GXN) to achieve comprehensive feature learning from multiple scales of a graph. Based on trainable hierarchical representations of a graph, GXN enables the interchange of intermediate features across scales to promote information flow. Two key ingredients of GXN include a novel vertex infomax pooling (VIPool), which creates multiscale graphs in a trainable manner, and a novel feature-crossing layer, enabling feature interchange across scales. The proposed VIPool selects the most informative subset of vertices based on the neural estimation of mutual information between vertex features and neighborhood features. The intuition behind is that a vertex is informative when it can maximally reflect its neighboring information. The proposed feature-crossing layer fuses intermediate features between two scales for mutual enhancement by improving information flow and enriching multiscale features at hidden layers. The cross shape of the feature-crossing layer distinguishes GXN from many other multiscale architectures. Experimental results show that the proposed GXN improves the classification accuracy by 2.12% and 1.15% on average for graph classification and vertex classification, respectively. Based on the same network, the proposed VIPool consistently outperforms other graph-pooling methods.

Imitation learning in a high-dimensional environment is challenging. Most inverse reinforcement learning (IRL) methods fail to outperform the demonstrator in such a high-dimensional environment, e.g., Atari domain. To address this challenge, we propose a novel reward learning module to generate intrinsic reward signals via a generative model. Our generative method can perform better forward state transition and backward action encoding, which improves the module's dynamics modeling ability in the environment. Thus, our module provides the imitation agent both the intrinsic intention of the demonstrator and a better exploration ability, which is critical for the agent to outperform the demonstrator. Empirical results show that our method outperforms state-of-the-art IRL methods on multiple Atari games, even with one-life demonstration. Remarkably, our method achieves performance that is up to 5 times the performance of the demonstration.

Multi-view alignment, achieving one-to-one correspondence of multi-view inputs, is critical in many real-world multi-view applications, especially for cross-view data analysis problems. Recently, an increasing number of works study this alignment problem with Canonical Correlation Analysis (CCA). However, existing CCA models are prone to misalign the multiple views due to either the neglect of uncertainty or the inconsistent encoding of the multiple views. To tackle these two issues, this paper studies multi-view alignment from the Bayesian perspective. Delving into the impairments of inconsistent encodings, we propose to recover correspondence of the multi-view inputs by matching the marginalization of the joint distribution of multi-view random variables under different forms of factorization. To realize our design, we present Adversarial CCA (ACCA) which achieves consistent latent encodings by matching the marginalized latent encodings through the adversarial training paradigm. Our analysis based on conditional mutual information reveals that ACCA is flexible for handling implicit distributions. Extensive experiments on correlation analysis and cross-view generation under noisy input settings demonstrate the superiority of our model., Comment: 37 pages, 22 figures

Distance Metric Learning (DML) has drawn much attention over the last two decades. A number of previous works have shown that it performs well in measuring the similarities of individuals given a set of correctly labeled pairwise data by domain experts. These important and precisely-labeled pairwise data are often highly sensitive in real world (e.g., patients similarity). This paper studies, for the first time, how pairwise information can be leaked to attackers during distance metric learning, and develops differential pairwise privacy (DPP), generalizing the definition of standard differential privacy, for secure metric learning. Unlike traditional differential privacy which only applies to independent samples, thus cannot be used for pairwise data, DPP successfully deals with this problem by reformulating the worst case. Specifically, given the pairwise data, we reveal all the involved correlations among pairs in the constructed undirected graph. DPP is then formalized that defines what kind of DML algorithm is private to preserve pairwise data. After that, a case study employing the contrastive loss is exhibited to clarify the details of implementing a DPP-DML algorithm. Particularly, the sensitivity reduction technique is proposed to enhance the utility of the output distance metric. Experiments both on a toy dataset and benchmarks demonstrate that the proposed scheme achieves pairwise data privacy without compromising the output performance much (Accuracy declines less than 0.01 throughout all benchmark datasets when the privacy budget is set at 4).

Black-box optimization is primarily important for many compute-intensive applications, including reinforcement learning (RL), robot control, etc. This paper presents a novel theoretical framework for black-box optimization, in which our method performs stochastic update with the implicit natural gradient of an exponential-family distribution. Theoretically, we prove the convergence rate of our framework with full matrix update for convex functions. Our theoretical results also hold for continuous non-differentiable black-box functions. Our methods are very simple and contain less hyper-parameters than CMA-ES \cite{hansen2006cma}. Empirically, our method with full matrix update achieves competitive performance compared with one of the state-of-the-art method CMA-ES on benchmark test problems. Moreover, our methods can achieve high optimization precision on some challenging test functions (e.g., $l_1$-norm ellipsoid test problem and Levy test problem), while methods with explicit natural gradient, i.e., IGO \cite{ollivier2017information} with full matrix update can not. This shows the efficiency of our methods., Comment: Black-box Optimization

Graph structured data provide two-fold information: graph structures and node attributes. Numerous graph-based algorithms rely on both information to achieve success in supervised tasks, such as node classification and link prediction. However, node attributes could be missing or incomplete, which significantly deteriorates the performance. The task of node attribute generation aims to generate attributes for those nodes whose attributes are completely unobserved. This task benefits many real-world problems like profiling, node classification and graph data augmentation. To tackle this task, we propose a deep adversarial learning based method to generate node attributes; called node attribute neural generator (NANG). NANG learns a unifying latent representation which is shared by both node attributes and graph structures and can be translated to different modalities. We thus use this latent representation as a bridge to convert information from one modality to another. We further introduce practical applications to quantify the performance of node attribute generation. Extensive experiments are conducted on four real-world datasets and the empirical results show that node attributes generated by the proposed method are high-qualitative and beneficial to other applications. The datasets and codes are available online.

In rank aggregation (RA), a collection of preferences from different users are summarized into a total order under the assumption of homogeneity of users. Model misspecification in RA arises since the homogeneity assumption fails to be satisfied in the complex real-world situation. Existing robust RAs usually resort to an augmentation of the ranking model to account for additional noises, where the collected preferences can be treated as a noisy perturbation of idealized preferences. Since the majority of robust RAs rely on certain perturbation assumptions, they cannot generalize well to agnostic noise-corrupted preferences in the real world. In this paper, we propose CoarsenRank, which possesses robustness against model misspecification. Specifically, the properties of our CoarsenRank are summarized as follows: (1) CoarsenRank is designed for mild model misspecification, which assumes there exist the ideal preferences (consistent with model assumption) that locates in a neighborhood of the actual preferences. (2) CoarsenRank then performs regular RAs over a neighborhood of the preferences instead of the original dataset directly. Therefore, CoarsenRank enjoys robustness against model misspecification within a neighborhood. (3) The neighborhood of the dataset is defined via their empirical data distributions. Further, we put an exponential prior on the unknown size of the neighborhood, and derive a much-simplified posterior formula for CoarsenRank under particular divergence measures. (4) CoarsenRank is further instantiated to Coarsened Thurstone, Coarsened Bradly-Terry, and Coarsened Plackett-Luce with three popular probability ranking models. Meanwhile, tractable optimization strategies are introduced with regards to each instantiation respectively. In the end, we apply CoarsenRank on four real-world datasets., Comment: https://github.com/Numb3rs-UTS/Robust-Ranking-Aggregation--CoarsenRank

In this work, we investigate black-box optimization from the perspective of frequentist kernel methods. We propose a novel batch optimization algorithm, which jointly maximizes the acquisition function and select points from a whole batch in a holistic way. Theoretically, we derive regret bounds for both the noise-free and perturbation settings irrespective of the choice of kernel. Moreover, we analyze the property of the adversarial regret that is required by a robust initialization for Bayesian Optimization (BO). We prove that the adversarial regret bounds decrease with the decrease of covering radius, which provides a criterion for generating a point set to minimize the bound. We then propose fast searching algorithms to generate a point set with a small covering radius for the robust initialization. Experimental results on both synthetic benchmark problems and real-world problems show the effectiveness of the proposed algorithms.

Deep neural networks (DNNs) have great expressive power, which can even memorize samples with wrong labels. It is vitally important to reiterate robustness and generalization in DNNs against label corruption. To this end, this paper studies the 0-1 loss, which has a monotonic relationship with an empirical adversary (reweighted) risk~\citep{hu2016does}. Although the 0-1 loss has some robust properties, it is difficult to optimize. To efficiently optimize the 0-1 loss while keeping its robust properties, we propose a very simple and efficient loss, i.e. curriculum loss (CL). Our CL is a tighter upper bound of the 0-1 loss compared with conventional summation based surrogate losses. Moreover, CL can adaptively select samples for model training. As a result, our loss can be deemed as a novel perspective of curriculum sample selection strategy, which bridges a connection between curriculum learning and robust learning. Experimental results on benchmark datasets validate the robustness of the proposed loss., Comment: ICLR2020

Learning with noisy labels, which aims to reduce expensive labors on accurate annotations, has become imperative in the Big Data era. Previous noise transition based method has achieved promising results and presented a theoretical guarantee on performance in the case of class-conditional noise. However, this type of approaches critically depend on an accurate pre-estimation of the noise transition, which is usually impractical. Subsequent improvement adapts the pre-estimation along with the training progress via a Softmax layer. However, the parameters in the Softmax layer are highly tweaked for the fragile performance due to the ill-posed stochastic approximation. To address these issues, we propose a Latent Class-Conditional Noise model (LCCN) that naturally embeds the noise transition under a Bayesian framework. By projecting the noise transition into a Dirichlet-distributed space, the learning is constrained on a simplex based on the whole dataset, instead of some ad-hoc parametric space. We then deduce a dynamic label regression method for LCCN to iteratively infer the latent labels, to stochastically train the classifier and to model the noise. Our approach safeguards the bounded update of the noise transition, which avoids previous arbitrarily tuning via a batch of samples. We further generalize LCCN for open-set noisy labels and the semi-supervised setting. We perform extensive experiments with the controllable noise data sets, CIFAR-10 and CIFAR-100, and the agnostic noise data sets, Clothing1M and WebVision17. The experimental results have demonstrated that the proposed model outperforms several state-of-the-art methods., Comment: New complete and extended version will be updated in the future

Learning with noisy labels is one of the hottest problems in weakly-supervised learning. Based on memorization effects of deep neural networks, training on small-loss instances becomes very promising for handling noisy labels. This fosters the state-of-the-art approach "Co-teaching" that cross-trains two deep neural networks using the small-loss trick. However, with the increase of epochs, two networks converge to a consensus and Co-teaching reduces to the self-training MentorNet. To tackle this issue, we propose a robust learning paradigm called Co-teaching+, which bridges the "Update by Disagreement" strategy with the original Co-teaching. First, two networks feed forward and predict all data, but keep prediction disagreement data only. Then, among such disagreement data, each network selects its small-loss data, but back propagates the small-loss data from its peer network and updates its own parameters. Empirical results on benchmark datasets demonstrate that Co-teaching+ is much superior to many state-of-the-art methods in the robustness of trained models.

Multi-output learning aims to simultaneously predict multiple outputs given an input. It is an important learning problem due to the pressing need for sophisticated decision making in real-world applications. Inspired by big data, the 4Vs characteristics of multi-output imposes a set of challenges to multi-output learning, in terms of the volume, velocity, variety and veracity of the outputs. Increasing number of works in the literature have been devoted to the study of multi-output learning and the development of novel approaches for addressing the challenges encountered. However, it lacks a comprehensive overview on different types of challenges of multi-output learning brought by the characteristics of the multiple outputs and the techniques proposed to overcome the challenges. This paper thus attempts to fill in this gap to provide a comprehensive review on this area. We first introduce different stages of the life cycle of the output labels. Then we present the paradigm on multi-output learning, including its myriads of output structures, definitions of its different sub-problems, model evaluation metrics and popular data repositories used in the study. Subsequently, we review a number of state-of-the-art multi-output learning methods, which are categorized based on the challenges., Comment: Paper accepted by IEEE Transactions on Neural Networks and Learning Systems

It is challenging for stochastic optimizations to handle large-scale sensitive data safely. Recently, Duchi et al. proposed private sampling strategy to solve privacy leakage in stochastic optimizations. However, this strategy leads to robustness degeneration, since this strategy is equal to the noise injection on each gradient, which adversely affects updates of the primal variable. To address this challenge, we introduce a robust stochastic optimization under the framework of local privacy, which is called Privacy-pREserving StochasTIc Gradual lEarning (PRESTIGE). PRESTIGE bridges private updates of the primal variable (by private sampling) with the gradual curriculum learning (CL). Specifically, the noise injection leads to the issue of label noise, but the robust learning process of CL can combat with label noise. Thus, PRESTIGE yields "private but robust" updates of the primal variable on the private curriculum, namely an reordered label sequence provided by CL. In theory, we reveal the convergence rate and maximum complexity of PRESTIGE. Empirical results on six datasets show that, PRESTIGE achieves a good tradeoff between privacy preservation and robustness over baselines.

To reduce the label complexity in Agnostic Active Learning (A^2 algorithm), volume-splitting splits the hypothesis edges to reduce the Vapnik-Chervonenkis (VC) dimension in version space. However, the effectiveness of volume-splitting critically depends on the initial hypothesis and this problem is also known as target-dependent label complexity gap. This paper attempts to minimize this gap by introducing a novel notion of number density which provides a more natural and direct way to describe the hypothesis distribution than volume. By discovering the connections between hypothesis and input distribution, we map the volume of version space into the number density and propose a target-independent distribution-splitting strategy with the following advantages: 1) provide theoretical guarantees on reducing label complexity and error rate as volume-splitting; 2) break the curse of initial hypothesis; 3) provide model guidance for a target-independent AL algorithm in real AL tasks. With these guarantees, for AL application, we then split the input distribution into more near-optimal spheres and develop an application algorithm called Distribution-based A^2 (DA^2). Experiments further verify the effectiveness of the halving and querying abilities of DA^2. Contributions of this paper are as follows., Comment: This paper has been withdrawn. The first author quitted the PhD study from AAI, University of Technology Sydney. The manuscript stopped updating

In this paper, we study two challenging problems in explainable AI (XAI) and data clustering. The first is how to directly design a neural network with inherent interpretability, rather than giving post-hoc explanations of a black-box model. The second is implementing discrete $k$-means with a differentiable neural network that embraces the advantages of parallel computing, online clustering, and clustering-favorable representation learning. To address these two challenges, we design a novel neural network, which is a differentiable reformulation of the vanilla $k$-means, called inTerpretable nEuraL cLustering (TELL). Our contributions are threefold. First, to the best of our knowledge, most existing XAI works focus on supervised learning paradigms. This work is one of the few XAI studies on unsupervised learning, in particular, data clustering. Second, TELL is an interpretable, or the so-called intrinsically explainable and transparent model. In contrast, most existing XAI studies resort to various means for understanding a black-box model with post-hoc explanations. Third, from the view of data clustering, TELL possesses many properties highly desired by $k$-means, including but not limited to online clustering, plug-and-play module, parallel computing, and provable convergence. Extensive experiments show that our method achieves superior performance comparing with 14 clustering approaches on three challenging data sets. The source code could be accessed at \url{www.pengxi.me}., Comment: 28 pages

Active Learning (AL) is a learning task that requires learners interactively query the labels of the sampled unlabeled instances to minimize the training outputs with human supervisions. In theoretical study, learners approximate the version space which covers all possible classification hypothesis into a bounded convex body and try to shrink the volume of it into a half-space by a given cut size. However, only the hypersphere with finite VC dimensions has obtained formal approximation guarantees that hold when the classes of Euclidean space are separable with a margin. In this paper, we approximate the version space to a structured {hypersphere} that covers most of the hypotheses, and then divide the available AL sampling approaches into two kinds of strategies: Outer Volume Sampling and Inner Volume Sampling. After providing provable guarantees for the performance of AL in version space, we aggregate the two kinds of volumes to eliminate their sampling biases via finding the optimal inscribed hyperspheres in the enclosing space of outer volume. To touch the version space from Euclidean space, we propose a theoretical bridge called Volume-based Model that increases the `sampling target-independent'. In non-linear feature space, spanned by kernel, we use sequential optimization to globally optimize the original space to a sparse space by halving the size of the kernel space. Then, the EM (Expectation Maximization) model which returns the local center helps us to find a local representation. To describe this process, we propose an easy-to-implement algorithm called Volume-based AL (VAL)., Comment: This paper has been withdrawn. The first author quitted the PhD study from AAI, University of Technology Sydney. The manuscript stopped updating

In information theory, Fisher information and Shannon information (entropy) are respectively used to quantify the uncertainty associated with the distribution modeling and the uncertainty in specifying the outcome of given variables. These two quantities are complementary and are jointly applied to information behavior analysis in most cases. The uncertainty property in information asserts a fundamental trade-off between Fisher information and Shannon information, which enlightens us the relationship between the encoder and the decoder in variational auto-encoders (VAEs). In this paper, we investigate VAEs in the Fisher-Shannon plane and demonstrate that the representation learning and the log-likelihood estimation are intrinsically related to these two information quantities. Through extensive qualitative and quantitative experiments, we provide with a better comprehension of VAEs in tasks such as high-resolution reconstruction, and representation learning in the perspective of Fisher information and Shannon information. We further propose a variant of VAEs, termed as Fisher auto-encoder (FAE), for practical needs to balance Fisher information and Shannon information. Our experimental results have demonstrated its promise in improving the reconstruction accuracy and avoiding the non-informative latent code as occurred in previous works.

We consider a challenging multi-label classification problem where both feature matrix $\X$ and label matrix $\Y$ have missing entries. An existing method concatenated $\X$ and $\Y$ as $[\X; \Y]$ and applied a matrix completion (MC) method to fill the missing entries, under the assumption that $[\X; \Y]$ is of low-rank. However, since entries of $\Y$ take binary values in the multi-label setting, it is unlikely that $\Y$ is of low-rank. Moreover, such assumption implies a linear relationship between $\X$ and $\Y$ which may not hold in practice. In this paper, we consider a latent matrix $\Z$ that produces the probability $\sigma(Z_{ij})$ of generating label $Y_{ij}$, where $\sigma(\cdot)$ is nonlinear. Considering label correlation, we assume $[\X; \Z]$ is of low-rank, and propose an MC algorithm based on subgradient descent named co-completion (COCO) motivated by elastic net and one-bit MC. We give a theoretical bound on the recovery effect of COCO and demonstrate its practical usefulness through experiments.

In this paper, we propose a simple variant of the original SVRG, called variance reduced stochastic gradient descent (VR-SGD). Unlike the choices of snapshot and starting points in SVRG and its proximal variant, Prox-SVRG, the two vectors of VR-SGD are set to the average and last iterate of the previous epoch, respectively. The settings allow us to use much larger learning rates, and also make our convergence analysis more challenging. We also design two different update rules for smooth and non-smooth objective functions, respectively, which means that VR-SGD can tackle non-smooth and/or non-strongly convex problems directly without any reduction techniques. Moreover, we analyze the convergence properties of VR-SGD for strongly convex problems, which show that VR-SGD attains linear convergence. Different from its counterparts that have no convergence guarantees for non-strongly convex problems, we also provide the convergence guarantees of VR-SGD for this case, and empirically verify that VR-SGD with varying learning rates achieves similar performance to its momentum accelerated variant that has the optimal convergence rate $\mathcal{O}(1/T^2)$. Finally, we apply VR-SGD to solve various machine learning problems, such as convex and non-convex empirical risk minimization, and leading eigenvalue computation. Experimental results show that VR-SGD converges significantly faster than SVRG and Prox-SVRG, and usually outperforms state-of-the-art accelerated methods, e.g., Katyusha., Comment: 46 pages, 25 figures. IEEE Transactions on Knowledge and Data Engineering, accepted in October, 2018. arXiv admin note: substantial text overlap with arXiv:1704.04966

While enormous progress has been made to Variational Autoencoder (VAE) in recent years, similar to other deep networks, VAE with deep networks suffers from the problem of degeneration, which seriously weakens the correlation between the input and the corresponding latent codes, deviating from the goal of the representation learning. To investigate how degeneration affects VAE from a theoretical perspective, we illustrate the information transmission in VAE and analyze the intermediate layers of the encoders/decoders. Specifically, we propose a Fisher Information measure for the layer-wise analysis. With such measure, we demonstrate that information loss is ineluctable in feed-forward networks and causes the degeneration in VAE. We show that skip connections in VAE enable the preservation of information without changing the model architecture. We call this class of VAE equipped with skip connections as SCVAE and perform a range of experiments to show its advantages in information preservation and degeneration mitigation.

Most existing learning to hash methods assume that there are sufficient data, either labeled or unlabeled, on the domain of interest (i.e., the target domain) for training. However, this assumption cannot be satisfied in some real-world applications. To address this data sparsity issue in hashing, inspired by transfer learning, we propose a new framework named Transfer Hashing with Privileged Information (THPI). Specifically, we extend the standard learning to hash method, Iterative Quantization (ITQ), in a transfer learning manner, namely ITQ+. In ITQ+, a new slack function is learned from auxiliary data to approximate the quantization error in ITQ. We developed an alternating optimization approach to solve the resultant optimization problem for ITQ+. We further extend ITQ+ to LapITQ+ by utilizing the geometry structure among the auxiliary data for learning more precise binary codes in the target domain. Extensive experiments on several benchmark datasets verify the effectiveness of our proposed approaches through comparisons with several state-of-the-art baselines., Comment: Accepted by IJCAI-2016

Many scientific datasets are of high dimension, and the analysis usually requires visual manipulation by retaining the most important structures of data. Principal curve is a widely used approach for this purpose. However, many existing methods work only for data with structures that are not self-intersected, which is quite restrictive for real applications. A few methods can overcome the above problem, but they either require complicated human-made rules for a specific task with lack of convergence guarantee and adaption flexibility to different tasks, or cannot obtain explicit structures of data. To address these issues, we develop a new regularized principal graph learning framework that captures the local information of the underlying graph structure based on reversed graph embedding. As showcases, models that can learn a spanning tree or a weighted undirected $\ell_1$ graph are proposed, and a new learning algorithm is developed that learns a set of principal points and a graph structure from data, simultaneously. The new algorithm is simple with guaranteed convergence. We then extend the proposed framework to deal with large-scale data. Experimental results on various synthetic and six real world datasets show that the proposed method compares favorably with baselines and can uncover the underlying structure correctly.

Matching Pursuit LASSIn Part I \cite{TanPMLPart1}, a Matching Pursuit LASSO ({MPL}) algorithm has been presented for solving large-scale sparse recovery (SR) problems. In this paper, we present a subspace search to further improve the performance of MPL, and then continue to address another major challenge of SR -- batch SR with many signals, a consideration which is absent from most of previous $\ell_1$-norm methods. As a result, a batch-mode {MPL} is developed to vastly speed up sparse recovery of many signals simultaneously. Comprehensive numerical experiments on compressive sensing and face recognition tasks demonstrate the superior performance of MPL and BMPL over other methods considered in this paper, in terms of sparse recovery ability and efficiency. In particular, BMPL is up to 400 times faster than existing $\ell_1$-norm methods considered to be state-of-the-art.O Part II: Applications and Sparse Recovery over Batch Signals

Due to the growing ubiquity of unlabeled data, learning with unlabeled data is attracting increasing attention in machine learning. In this paper, we propose a novel semi-supervised kernel learning method which can seamlessly combine manifold structure of unlabeled data and Regularized Least-Squares (RLS) to learn a new kernel. Interestingly, the new kernel matrix can be obtained analytically with the use of spectral decomposition of graph Laplacian matrix. Hence, the proposed algorithm does not require any numerical optimization solvers. Moreover, by maximizing kernel target alignment on labeled data, we can also learn model parameters automatically with a closed-form solution. For a given graph Laplacian matrix, our proposed method does not need to tune any model parameter including the tradeoff parameter in RLS and the balance parameter for unlabeled data. Extensive experiments on ten benchmark datasets show that our proposed two-stage parameter-free spectral kernel learning algorithm can obtain comparable performance with fine-tuned manifold regularization methods in transductive setting, and outperform multiple kernel learning in supervised setting., Comment: Appears in Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence (UAI2010)

Due to myriads of classes, designing accurate and efficient classifiers becomes very challenging for multi-class classification. Recent research has shown that class structure learning can greatly facilitate multi-class learning. In this paper, we propose a novel method to learn the class structure for multi-class classification problems. The class structure is assumed to be a binary hierarchical tree. To learn such a tree, we propose a maximum separating margin method to determine the child nodes of any internal node. The proposed method ensures that two classgroups represented by any two sibling nodes are most separable. In the experiments, we evaluate the accuracy and efficiency of the proposed method over other multi-class classification methods on real world large-scale problems. The results show that the proposed method outperforms benchmark methods in terms of accuracy for most datasets and performs comparably with other class structure learning methods in terms of efficiency for all datasets.