We developed a Nonlinear Level-set Learning (NLL) method for dimensionality reduction in high-dimensional function approximation with small data. This work is motivated by a variety of design tasks in real-world engineering applications, where practitioners would replace their computationally intensive physical models (e.g., high-resolution fluid simulators) with fast-to-evaluate predictive machine learning models, so as to accelerate the engineering design processes. There are two major challenges in constructing such predictive models: (a) high-dimensional inputs (e.g., many independent design parameters) and (b) small training data, generated by running extremely time-consuming simulations. Thus, reducing the input dimension is critical to alleviate the over-fitting issue caused by data insufficiency. Existing methods, including sliced inverse regression and active subspace approaches, reduce the input dimension by learning a linear coordinate transformation; our main contribution is to extend the transformation approach to a nonlinear regime.

Dimensionality reduction plays a central role in real-world applications for Machine Learning, among many fields. In particular, metric dimensionality reduction where data from a general metric is mapped into low dimensional space, is often used as a first step before applying machine learning algorithms. In almost all these applications the quality of the embedding is measured by various average case criteria. Metric dimensionality reduction has also been studied in Math and TCS, within the extremely fruitful and influential field of metric embedding. Yet, the vast majority of theoretical research has been devoted to analyzing the worst case behavior of embeddings and therefore has little relevance to practical settings.

We revisit the classic randomized sketch of a tensor product of $q$ vectors $x_i\in\mathbb{R} n$. However, in their analysis $C_{\Omega} 2$ can be as large as $\Theta(n {2q})$, even for a set $\Omega$ of $O(1)$ vectors $x$. We give a new analysis of this sketch, providing nearly optimal bounds. For the important case of $q 2$ and $\delta 1/\poly(n)$, this shows that $m \Theta(\epsilon {-2} \log(n) \epsilon {-1} \log 2(n))$, demonstrating that the $\epsilon {-2}$ and $\log 2(n)$ terms do not multiply each other. In a number of applications, one has $ \Omega \poly(n)$ and in this case our bounds are optimal up to a constant factor.

Kernel dimensionality reduction (KDR) algorithms find a low dimensional representation of the original data by optimizing kernel dependency measures that are capable of capturing nonlinear relationships. The standard strategy is to first map the data into a high dimensional feature space using kernels prior to a projection onto a low dimensional space. While KDR methods can be easily solved by keeping the most dominant eigenvectors of the kernel matrix, its features are no longer easy to interpret. Alternatively, Interpretable KDR (IKDR) is different in that it projects onto a subspace \textit{before} the kernel feature mapping, therefore, the projection matrix can indicate how the original features linearly combine to form the new features. Unfortunately, the IKDR objective requires a non-convex manifold optimization that is difficult to solve and can no longer be solved by eigendecomposition.

This work develops an analytically solvable unsupervised learning scheme that extracts the most informative components for predicting future inputs, termed predictive principal component analysis (PredPCA). Our scheme can effectively remove unpredictable observation noise and globally minimize the test prediction error. Mathematical analyses demonstrate that, with sufficiently high-dimensional observations that are generated by a linear or nonlinear system, PredPCA can identify the optimal hidden state representation, true system parameters, and true hidden state dimensionality, with a global convergence guarantee. We demonstrate the performance of PredPCA by using sequential visual inputs comprising hand-digits, rotating 3D objects, and natural scenes. It reliably and accurately estimates distinct hidden states and predicts future outcomes of previously unseen test input data, even in the presence of considerable observation noise. The simple model structure and low computational cost of PredPCA make it highly desirable as a learning scheme for biological neural networks and neuromorphic chips. Prediction is essential for both biological organisms [1,2] and machine learning [3,4]. In particular, they need to predict the dynamics of newly encountered sensory input data (i.e., test data) based on and only on knowledge learned from a limited number of past experiences (i.e., training data). Generalization error is a standard measure of the generalization capability of predicting the future consequences of previously unseen input data, which is defined as the difference between the training and test prediction errors.

Movement primitives are an important policy class for real-world robotics. However, the high dimensionality of their parametrization makes the policy optimization expensive both in terms of samples and computation. Enabling an efficient representation of movement primitives facilitates the application of machine learning techniques such as reinforcement on robotics. Motions, especially in highly redundant kinematic structures, exhibit high correlation in the configuration space. For these reasons, prior work has mainly focused on the application of dimensionality reduction techniques in the configuration space. In this paper, we investigate the application of dimensionality reduction in the parameter space, identifying principal movements. The resulting approach is enriched with a probabilistic treatment of the parameters, inheriting all the properties of the Probabilistic Movement Primitives. We test the proposed technique both on a real robotic task and on a database of complex human movements. The empirical analysis shows that the dimensionality reduction in parameter space is more effective than in configuration space, as it enables the representation of the movements with a significant reduction of parameters.

The wide variety of motions performed by the human arm during daily tasks makes it desirable to find representative subsets to reduce the dimensionality of these movements for a variety of applications, including the design and control of robotic and prosthetic devices. This paper presents a novel method and the results of an extensive human subjects study to obtain representative arm joint angle trajectories that span naturalistic motions during Activities of Daily Living (ADLs). In particular, we seek to identify sets of useful motion trajectories of the upper limb that are functions of a single variable, allowing, for instance, an entire prosthetic or robotic arm to be controlled with a single input from a user, along with a means to select between motions for different tasks. Data driven approaches are used to obtain clusters as well as representative motion averages for the full-arm 7 degree of freedom (DOF), elbow-wrist 4 DOF, and wrist-only 3 DOF motions. The proposed method makes use of well-known techniques such as dynamic time warping (DTW) to obtain a divergence measure between motion segments, DTW barycenter averaging (DBA) to obtain averages, Ward's distance criterion to build hierarchical trees, batch-DTW to simultaneously align multiple motion data, and functional principal component analysis (fPCA) to evaluate cluster variability. The clusters that emerge associate various recorded motions into primarily hand start and end location for the full-arm system, motion direction for the wrist-only system, and an intermediate between the two qualities for the elbow-wrist system. The proposed clustering methodology is justified by comparing results against alternative approaches.

This paper introduces a new approach to constructing meaningful lower dimensional representations of sets of data points. We argue that constraining the mapping between the high and low dimensional spaces to be a diffeomorphism is a natural way of ensuring that pairwise distances are approximately preserved. Accordingly we develop an algorithm which diffeomorphically maps the data near to a lower dimensional subspace and then projects onto that subspace. The problem of solving for the mapping is transformed into one of solving for an Eulerian flow field which we compute using ideas from kernel methods. We demonstrate the efficacy of our approach on various real world data sets.

In solving complex visual learning tasks, adopting multiple descriptors to more precisely characterize the data has been a feasible way for improving performance. These representations are typically high dimensional and assume diverse forms. Thus finding a way to transform them into a unified space of lower dimension generally facilitates the underlying tasks, such as object recognition or clustering. We describe an approach that incorporates multiple kernel learning with dimensionality reduction (MKL-DR). While the proposed framework is flexible in simultaneously tackling data in various feature representations, the formulation itself is general in that it is established upon graph embedding. It follows that any dimensionality reduction techniques explainable by graph embedding can be generalized by our method to consider data in multiple feature representations.

Semi-supervised regression based on the graph Laplacian suffers from the fact that the solution is biased towards a constant and the lack of extrapolating power. Outgoing from these observations we propose to use the second-order Hessian energy for semi-supervised regression which overcomes both of these problems, in particular, if the data lies on or close to a low-dimensional submanifold in the feature space, the Hessian energy prefers functions which vary linearly with respect to the natural parameters in the data. This property makes it also particularly suited for the task of semi-supervised dimensionality reduction where the goal is to find the natural parameters in the data based on a few labeled points. The experimental result suggest that our method is superior to semi-supervised regression using Laplacian regularization and standard supervised methods and is particularly suited for semi-supervised dimensionality reduction. Papers published at the Neural Information Processing Systems Conference.