A Category Space Approach to Supervised Dimensionality Reduction

Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by the classes (or categories). To this end, we set up a model in which each class is represented as a 1D subspace of the vector space formed by the features. Assuming the set of classes does not exceed the cardinality of the features, the model results in multi-class supervised learning in which the features of each class are projected into the class subspace. Class discrimination is automatically guaranteed via the imposition of orthogonality of the 1D class sub-spaces. The resulting optimization problem - formulated as the minimization of a sum of quadratic functions on a Stiefel manifold - while being non-convex (due to the constraints), nevertheless has a structure for which we can identify when we have reached a global minimum. After formulating a version with standard inner products, we extend the formulation to reproducing kernel Hilbert spaces in a straightforward manner. The optimization approach also extends in a similar fashion to the kernel version. Results and comparisons with the multi-class Fisher linear (and kernel) discriminants and principal component analysis (linear and kernel) showcase the relative merits of this approach to dimensionality reduction.

A Compositional Approach to Language Modeling

Traditional language models treat language as a finite state automaton on a probability space over words. This is a very strong assumption when modeling something inherently complex such as language. In this paper, we challenge this by showing how the linear chain assumption inherent in previous work can be translated into a sequential composition tree. We then propose a new model that marginalizes over all possible composition trees thereby removing any underlying structural assumptions. As the partition function of this new model is intractable, we use a recently proposed sentence level evaluation metric Contrastive Entropy to evaluate our model. Given this new evaluation metric, we report more than 100% improvement across distortion levels over current state of the art recurrent neural network based language models.

Contrastive Entropy: A new evaluation metric for unnormalized language models

Perplexity (per word) is the most widely used metric for evaluating language models. Despite this, there has been no dearth of criticism for this metric. Most of these criticisms center around lack of correlation with extrinsic metrics like word error rate (WER), dependence upon shared vocabulary for model comparison and unsuitability for unnormalized language model evaluation. In this paper, we address the last problem and propose a new discriminative entropy based intrinsic metric that works for both traditional word level models and unnormalized language models like sentence level models. We also propose a discriminatively trained sentence level interpretation of recurrent neural network based language model (RNN) as an example of unnormalized sentence level model. We demonstrate that for word level models, contrastive entropy shows a strong correlation with perplexity. We also observe that when trained at lower distortion levels, sentence level RNN considerably outperforms traditional RNNs on this new metric.

Disaggregation of SMAP L3 Brightness Temperatures to 9km using Kernel Machines

In this study, a machine learning algorithm is used for disaggregation of SMAP brightness temperatures (T$_{\textrm{B}}$) from 36km to 9km. It uses image segmentation to cluster the study region based on meteorological and land cover similarity, followed by a support vector machine based regression that computes the value of the disaggregated T$_{\textrm{B}}$ at all pixels. High resolution remote sensing products such as land surface temperature, normalized difference vegetation index, enhanced vegetation index, precipitation, soil texture, and land-cover were used for disaggregation. The algorithm was implemented in Iowa, United States, from April to July 2015, and compared with the SMAP L3_SM_AP T$_{\textrm{B}}$ product at 9km. It was found that the disaggregated T$_{\textrm{B}}$ were very similar to the SMAP-T$_{\textrm{B}}$ product, even for vegetated areas with a mean difference $\leq$ 5K. However, the standard deviation of the disaggregation was lower by 7K than that of the AP product. The probability density functions of the disaggregated T$_{\textrm{B}}$ were similar to the SMAP-T$_{\textrm{B}}$. The results indicate that this algorithm may be used for disaggregating T$_{\textrm{B}}$ using complex non-linear correlations on a grid.

A Grassmannian Graph Approach to Affine Invariant Feature Matching

In this work, we present a novel and practical approach to address one of the longstanding problems in computer vision: 2D and 3D affine invariant feature matching. Our Grassmannian Graph (GrassGraph) framework employs a two stage procedure that is capable of robustly recovering correspondences between two unorganized, affinely related feature (point) sets. The first stage maps the feature sets to an affine invariant Grassmannian representation, where the features are mapped into the same subspace. It turns out that coordinate representations extracted from the Grassmannian differ by an arbitrary orthonormal matrix. In the second stage, by approximating the Laplace-Beltrami operator (LBO) on these coordinates, this extra orthonormal factor is nullified, providing true affine-invariant coordinates which we then utilize to recover correspondences via simple nearest neighbor relations. The resulting GrassGraph algorithm is empirically shown to work well in non-ideal scenarios with noise, outliers, and occlusions. Our validation benchmarks use an unprecedented 440,000+ experimental trials performed on 2D and 3D datasets, with a variety of parameter settings and competing methods. State-of-the-art performance in the majority of these extensive evaluations confirm the utility of our method.

Spatial Scaling of Satellite Soil Moisture using Temporal Correlations and Ensemble Learning

A novel algorithm is developed to downscale soil moisture (SM), obtained at satellite scales of 10-40 km by utilizing its temporal correlations to historical auxiliary data at finer scales. Including such correlations drastically reduces the size of the training set needed, accounts for time-lagged relationships, and enables downscaling even in the presence of short gaps in the auxiliary data. The algorithm is based upon bagged regression trees (BRT) and uses correlations between high-resolution remote sensing products and SM observations. The algorithm trains multiple regression trees and automatically chooses the trees that generate the best downscaled estimates. The algorithm was evaluated using a multi-scale synthetic dataset in north central Florida for two years, including two growing seasons of corn and one growing season of cotton per year. The time-averaged error across the region was found to be 0.01 $\mathrm{m}^3/\mathrm{m}^3$, with a standard deviation of 0.012 $\mathrm{m}^3/\mathrm{m}^3$ when 0.02% of the data were used for training in addition to temporal correlations from the past seven days, and all available data from the past year. The maximum spatially averaged errors obtained using this algorithm in downscaled SM were 0.005 $\mathrm{m}^3/\mathrm{m}^3$, for pixels with cotton land-cover. When land surface temperature~(LST) on the day of downscaling was not included in the algorithm to simulate "data gaps", the spatially averaged error increased minimally by 0.015 $\mathrm{m}^3/\mathrm{m}^3$ when LST is unavailable on the day of downscaling. The results indicate that the BRT-based algorithm provides high accuracy for downscaling SM using complex non-linear spatio-temporal correlations, under heterogeneous micro meteorological conditions.

Disaggregation of Remotely Sensed Soil Moisture in Heterogeneous Landscapes using Holistic Structure based Models

In this study, a novel machine learning algorithm is presented for disaggregation of satellite soil moisture (SM) based on self-regularized regressive models (SRRM) using high-resolution correlated information from auxiliary sources. It includes regularized clustering that assigns soft memberships to each pixel at fine-scale followed by a kernel regression that computes the value of the desired variable at all pixels. Coarse-scale remotely sensed SM were disaggregated from 10km to 1km using land cover, precipitation, land surface temperature, leaf area index, and in-situ observations of SM. This algorithm was evaluated using multi-scale synthetic observations in NC Florida for heterogeneous agricultural land covers. It was found that the root mean square error (RMSE) for 96% of the pixels was less than 0.02 $m^3/m^3$. The clusters generated represented the data well and reduced the RMSE by upto 40% during periods of high heterogeneity in land-cover and meteorological conditions. The Kullback Leibler divergence (KLD) between the true SM and the disaggregated estimates is close to 0, for both vegetated and baresoil landcovers. The disaggregated estimates were compared to those generated by the Principle of Relevant Information (PRI) method. The RMSE for the PRI disaggregated estimates is higher than the RMSE for the SRRM on each day of the season. The KLD of the disaggregated estimates generated by the SRRM is at least four orders of magnitude lower than those for the PRI disaggregated estimates, while the computational time needed was reduced by three times. The results indicate that the SRRM can be used for disaggregating SM with complex non-linear correlations on a grid with high accuracy.

A spatial compositional model (SCM) for linear unmixing and endmember uncertainty estimation

The normal compositional model (NCM) has been extensively used in hyperspectral unmixing. However, most of the previous research has focused on estimation of endmembers and/or their variability. Also, little work has employed spatial information in NCM. In this paper, we show that NCM can be used for calculating the uncertainty of the estimated endmembers with spatial priors incorporated for better unmixing. This results in a spatial compositional model (SCM) which features (i) spatial priors that force neighboring abundances to be similar based on their pixel similarity and (ii) a posterior that is obtained from a likelihood model which does not assume pixel independence. The resulting algorithm turns out to be easy to implement and efficient to run. We compared SCM with current state-of-the-art algorithms on synthetic and real images. The results show that SCM can in the main provide more accurate endmembers and abundances. Moreover, the estimated uncertainty can serve as a prediction of endmember error under certain conditions.

A new variational principle for the Euclidean distance function: Linear approach to the non-linear eikonal problem

We present a fast convolution-based technique for computing an approximate, signed Euclidean distance function $S$ on a set of 2D and 3D grid locations. Instead of solving the non-linear, static Hamilton-Jacobi equation ($\|\nabla S\|=1$), our solution stems from first solving for a scalar field $\phi$ in a linear differential equation and then deriving the solution for $S$ by taking the negative logarithm. In other words, when $S$ and $\phi$ are related by $\phi = \exp \left(-\frac{S}{\tau} \right)$ and $\phi$ satisfies a specific linear differential equation corresponding to the extremum of a variational problem, we obtain the approximate Euclidean distance function $S = -\tau \log(\phi)$ which converges to the true solution in the limit as $\tau \rightarrow 0$. This is in sharp contrast to techniques like the fast marching and fast sweeping methods which directly solve the Hamilton-Jacobi equation by the Godunov upwind discretization scheme. Our linear formulation results in a closed-form solution to the approximate Euclidean distance function expressible as a discrete convolution, and hence efficiently computable using the fast Fourier transform (FFT). Our solution also circumvents the need for spatial discretization of the derivative operator. As $\tau\rightarrow0$ we show the convergence of our results to the true solution and also bound the error for a given value of $\tau$. The differentiability of our solution allows us to compute---using a set of convolutions---the first and second derivatives of the approximate distance function. In order to determine the sign of the distance function (defined to be positive inside a closed region and negative outside), we compute the winding number in 2D and the topological degree in 3D, whose computations can also be performed via fast convolutions. We demonstrate the efficacy of our method through a set of experimental results.

A fast eikonal equation solver using the Schrodinger wave equation

We use a Schr\"odinger wave equation formalism to solve the eikonal equation. In our framework, a solution to the eikonal equation is obtained in the limit as Planck's constant $\hbar$ (treated as a free parameter) tends to zero of the solution to the corresponding linear Schr\"odinger equation. The Schr\"odinger equation corresponding to the eikonal turns out to be a \emph{generalized, screened Poisson equation}. Despite being linear, it does not have a closed-form solution for arbitrary forcing functions. We present two different techniques to solve the screened Poisson equation. In the first approach we use a standard perturbation analysis approach to derive a new algorithm which is guaranteed to converge provided the forcing function is bounded and positive. The perturbation technique requires a sequence of discrete convolutions which can be performed in $O(N\log N)$ using the Fast Fourier Transform (FFT) where $N$ is the number of grid points. In the second method we discretize the linear Laplacian operator by the finite difference method leading to a sparse linear system of equations which can be solved using the plethora of sparse solvers. The eikonal solution is recovered from the exponent of the resultant scalar field. Our approach eliminates the need to explicitly construct viscosity solutions as customary with direct solutions to the eikonal. Since the linear equation is computed for a small but non-zero $\hbar$, the obtained solution is an approximation. Though our solution framework is applicable to the general class of eikonal problems, we detail specifics for the popular vision applications of shape-from-shading, vessel segmentation, and path planning.