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Register a new userPolyGAN: High-Order Polynomial Generators
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Generative Adversarial Networks (GANs) have become the gold standard when it comes to learning generative models for high-dimensional distributions. Since their advent, numerous variations of GANs have been introduced in the literature, primarily focusing on utilization of novel loss functions, optimization/regularization strategies and network architectures. In this paper, we turn our attention to the generator and investigate the use of high-order polynomials as an alternative class of universal function approximators. Concretely, we propose PolyGAN, where we model the data generator by means of a high-order polynomial whose unknown parameters are naturally represented by high-order tensors. We introduce two tensor decompositions that significantly reduce the number of parameters and show how they can be efficiently implemented by hierarchical neural networks that only employ linear/convolutional blocks. We exhibit for the first time that by using our approach a GAN generator can approximate the data distribution without using any activation functions. Thorough experimental evaluation on both synthetic and real data (images and 3D point clouds) demonstrates the merits of PolyGAN against the state of the art.
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We study the sample complexity of private synthetic data generation over an unbounded sized class of statistical queries, and show that any class that is privately proper PAC learnable admits a private synthetic data generator (perhaps non-efficient). Previous work on synthetic data generators focused on the case that the query class $mathcal{D}$ is finite and obtained sample complexity bounds that scale logarithmically with the size $|mathcal{D}|$. Here we construct a private synthetic data generator whose sample complexity is independent of the domain size, and we replace finiteness with the assumption that $mathcal{D}$ is privately PAC learnable (a formally weaker task, hence we obtain equivalence between the two tasks).
Triangular map is a recent construct in probability theory that allows one to transform any source probability density function to any target density function. Based on triangular maps, we propose a general framework for high-dimensional density estimation, by specifying one-dimensional transformations (equivalently conditional densities) and appropriate conditioner networks. This framework (a) reveals the commonalities and differences of existing autoregressive and flow based methods, (b) allows a unified understanding of the limitations and representation power of these recent approaches and, (c) motivates us to uncover a new Sum-of-Squares (SOS) flow that is interpretable, universal, and easy to train. We perform several synthetic experiments on various density geometries to demonstrate the benefits (and short-comings) of such transformations. SOS flows achieve competitive results in simulations and several real-world datasets.
Multi-view spectral clustering can effectively reveal the intrinsic cluster structure among data by performing clustering on the learned optimal embedding across views. Though demonstrating promising performance in various applications, most of existing methods usually linearly combine a group of pre-specified first-order Laplacian matrices to construct the optimal Laplacian matrix, which may result in limited representation capability and insufficient information exploitation. Also, storing and implementing complex operations on the $ntimes n$ Laplacian matrices incurs intensive storage and computation complexity. To address these issues, this paper first proposes a multi-view spectral clustering algorithm that learns a high-order optimal neighborhood Laplacian matrix, and then extends it to the late fusion version for accurate and efficient multi-view clustering. Specifically, our proposed algorithm generates the optimal Laplacian matrix by searching the neighborhood of the linear combination of both the first-order and high-order base Laplacian matrices simultaneously. By this way, the representative capacity of the learned optimal Laplacian matrix is enhanced, which is helpful to better utilize the hidden high-order connection information among data, leading to improved clustering performance. We design an efficient algorithm with proved convergence to solve the resultant optimization problem. Extensive experimental results on nine datasets demonstrate the superiority of our algorithm against state-of-the-art methods, which verifies the effectiveness and advantages of the proposed algorithm.
The problem of explaining the behavior of deep neural networks has recently gained a lot of attention. While several attribution methods have been proposed, most come without strong theoretical foundations, which raises questions about their reliability. On the other hand, the literature on cooperative game theory suggests Shapley values as a unique way of assigning relevance scores such that certain desirable properties are satisfied. Unfortunately, the exact evaluation of Shapley values is prohibitively expensive, exponential in the number of input features. In this work, by leveraging recent results on uncertainty propagation, we propose a novel, polynomial-time approximation of Shapley values in deep neural networks. We show that our method produces significantly better approximations of Shapley values than existing state-of-the-art attribution methods.
We study the complexity of training neural network models with one hidden nonlinear activation layer and an output weighted sum layer. We analyze Gradient Descent applied to learning a bounded target function on $n$ real-valued inputs. We give an agnostic learning guarantee for GD: starting from a randomly initialized network, it converges in mean squared loss to the minimum error (in $2$-norm) of the best approximation of the target function using a polynomial of degree at most $k$. Moreover, for any $k$, the size of the network and number of iterations needed are both bounded by $n^{O(k)}log(1/epsilon)$. In particular, this applies to training networks of unbiased sigmoids and ReLUs. We also rigorously explain the empirical finding that gradient descent discovers lower frequency Fourier components before higher frequency components. We complement this result with nearly matching lower bounds in the Statistical Query model. GD fits well in the SQ framework since each training step is determined by an expectation over the input distribution. We show that any SQ algorithm that achieves significant improvement over a constant function with queries of tolerance some inverse polynomial in the input dimensionality $n$ must use $n^{Omega(k)}$ queries even when the target functions are restricted to a set of $n^{O(k)}$ degree-$k$ polynomials, and the input distribution is uniform over the unit sphere; for this class the information-theoretic lower bound is only $Theta(k log n)$. Our approach for both parts is based on spherical harmonics. We view gradient descent as an operator on the space of functions, and study its dynamics. An essential tool is the Funk-Hecke theorem, which explains the eigenfunctions of this operator in the case of the mean squared loss.
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