Learning from Geometry

Subspaces and manifolds are two powerful models for high dimensional signals. Subspaces model linear correlation and are a good fit to signals generated by physical systems, such as frontal images of human faces and multiple sources impinging at an antenna array. Manifolds model sources that are not linearly correlated, but where signals are determined by a small number of parameters. Examples are images of human faces under different poses or expressions, and handwritten digits with varying styles. However, there will always be some degree of model mismatch between the subspace or manifold model and the true statistics of the source. This dissertation exploits subspace and manifold models as prior information in various signal processing and machine learning tasks.

A near-low-rank Gaussian mixture model measures proximity to a union of linear or affine subspaces. This simple model can effectively capture the signal distribution when each class is near a subspace. This dissertation studies how the pairwise geometry between these subspaces affects classification performance. When model mismatch is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the model mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. This linear transformation, termed TRAIT, also preserves some specific features in each class, being complementary to a recently developed Low Rank Transform (LRT). Moreover, when the model mismatch is more significant, TRAIT shows superior performance compared to LRT.

The manifold model enforces a constraint on the freedom of data variation. Learning features that are robust to data variation is very important, especially when the size of the training set is small. A learning machine with large numbers of parameters, e.g., deep neural network, can well describe a very complicated data distribution. However, it is also more likely to be sensitive to small perturbations of the data, and to suffer from suffer from degraded performance when generalizing to unseen (test) data.

From the perspective of complexity of function classes, such a learning machine has a huge capacity (complexity), which tends to overfit. The manifold model provides us with a way of regularizing the learning machine, so as to reduce the generalization error, therefore mitigate overfiting. Two different overfiting-preventing approaches are proposed, one from the perspective of data variation, the other from capacity/complexity control. In the first approach, the learning machine is encouraged to make decisions that vary smoothly for data points in local neighborhoods on the manifold. In the second approach, a graph adjacency matrix is derived for the manifold, and the learned features are encouraged to be aligned with the principal components of this adjacency matrix. Experimental results on benchmark datasets are demonstrated, showing an obvious advantage of the proposed approaches when the training set is small.

Stochastic optimization makes it possible to track a slowly varying subspace underlying streaming data. By approximating local neighborhoods using affine subspaces, a slowly varying manifold can be efficiently tracked as well, even with corrupted and noisy data. The more the local neighborhoods, the better the approximation, but the higher the computational complexity. A multiscale approximation scheme is proposed, where the local approximating subspaces are organized in a tree structure. Splitting and merging of the tree nodes then allows efficient control of the number of neighbourhoods. Deviation (of each datum) from the learned model is estimated, yielding a series of statistics for anomaly detection. This framework extends the classical {\em changepoint detection} technique, which only works for one dimensional signals. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.

Learning and Exploiting Camera Geometry for Computer Vision

This work explores the use of statistical methods in describing and estimating camera poses, as well as the information feedback loop between camera pose and object detection. Surging development in robotics and computer vision has pushed the need for algorithms that infer, understand, and utilize information about the position and orientation of the sensor platforms when observing and/or interacting with their environment.

The first contribution of this thesis is the development of a set of statistical tools for representing and estimating the uncertainty in object poses. A distribution for representing the joint uncertainty over multiple object positions and orientations is described, called the mirrored normal-Bingham distribution. This distribution generalizes both the normal distribution in Euclidean space, and the Bingham distribution on the unit hypersphere. It is shown to inherit many of the convenient properties of these special cases: it is the maximum-entropy distribution with fixed second moment, and there is a generalized Laplace approximation whose result is the mirrored normal-Bingham distribution. This distribution and approximation method are demonstrated by deriving the analytical approximation to the wrapped-normal distribution. Further, it is shown how these tools can be used to represent the uncertainty in the result of a bundle adjustment problem.

Another application of these methods is illustrated as part of a novel camera pose estimation algorithm based on object detections. The autocalibration task is formulated as a bundle adjustment problem using prior distributions over the 3D points to enforce the objects' structure and their relationship with the scene geometry. This framework is very flexible and enables the use of off-the-shelf computational tools to solve specialized autocalibration problems. Its performance is evaluated using a pedestrian detector to provide head and foot location observations, and it proves much faster and potentially more accurate than existing methods.

Finally, the information feedback loop between object detection and camera pose estimation is closed by utilizing camera pose information to improve object detection in scenarios with significant perspective warping. Methods are presented that allow the inverse perspective mapping traditionally applied to images to be applied instead to features computed from those images. For the special case of HOG-like features, which are used by many modern object detection systems, these methods are shown to provide substantial performance benefits over unadapted detectors while achieving real-time frame rates, orders of magnitude faster than comparable image warping methods.

The statistical tools and algorithms presented here are especially promising for mobile cameras, providing the ability to autocalibrate and adapt to the camera pose in real time. In addition, these methods have wide-ranging potential applications in diverse areas of computer vision, robotics, and imaging.

Measuring morphological features using light-scattering spectroscopy and Fourier-domain low-coherence interferometry.

We present measurements of morphological features in a thick turbid sample using light-scattering spectroscopy (LSS) and Fourier-domain low-coherence interferometry (fLCI) by processing with the dual-window (DW) method. A parallel frequency domain optical coherence tomography (OCT) system with a white-light source is used to image a two-layer phantom containing polystyrene beads of diameters 4.00 and 6.98 mum on the top and bottom layers, respectively. The DW method decomposes each OCT A-scan into a time-frequency distribution with simultaneously high spectral and spatial resolution. The spectral information from localized regions in the sample is used to determine scatterer structure. The results show that the two scatterer populations can be differentiated using LSS and fLCI.

Evolution of the sex-related locus and genomic features shared in microsporidia and fungi.

BACKGROUND: Microsporidia are obligate intracellular, eukaryotic pathogens that infect a wide range of animals from nematodes to humans, and in some cases, protists. The preponderance of evidence as to the origin of the microsporidia reveals a close relationship with the fungi, either within the kingdom or as a sister group to it. Recent phylogenetic studies and gene order analysis suggest that microsporidia share a particularly close evolutionary relationship with the zygomycetes. METHODOLOGY/PRINCIPAL FINDINGS: Here we expanded this analysis and also examined a putative sex-locus for variability between microsporidian populations. Whole genome inspection reveals a unique syntenic gene pair (RPS9-RPL21) present in the vast majority of fungi and the microsporidians but not in other eukaryotic lineages. Two other unique gene fusions (glutamyl-prolyl tRNA synthetase and ubiquitin-ribosomal subunit S30) that are present in metazoans, choanoflagellates, and filasterean opisthokonts are unfused in the fungi and microsporidians. One locus previously found to be conserved in many microsporidian genomes is similar to the sex locus of zygomycetes in gene order and architecture. Both sex-related and sex loci harbor TPT, HMG, and RNA helicase genes forming a syntenic gene cluster. We sequenced and analyzed the sex-related locus in 11 different Encephalitozoon cuniculi isolates and the sibling species E. intestinalis (3 isolates) and E. hellem (1 isolate). There was no evidence for an idiomorphic sex-related locus in this Encephalitozoon species sample. According to sequence-based phylogenetic analyses, the TPT and RNA helicase genes flanking the HMG genes are paralogous rather than orthologous between zygomycetes and microsporidians. CONCLUSION/SIGNIFICANCE: The unique genomic hallmarks between microsporidia and fungi are independent of sequence based phylogenetic comparisons and further contribute to define the borders of the fungal kingdom and support the classification of microsporidia as unusual derived fungi. And the sex/sex-related loci appear to have been subject to frequent gene conversion and translocations in microsporidia and zygomycetes.

Features of programmed cell death in intact Xenopus oocytes and early embryos revealed by near-infrared fluorescence and real-time monitoring.

Factors influencing apoptosis of vertebrate eggs and early embryos have been studied in cell-free systems and in intact embryos by analyzing individual apoptotic regulators or caspase activation in static samples. A novel method for monitoring caspase activity in living Xenopus oocytes and early embryos is described here. The approach, using microinjection of a near-infrared caspase substrate that emits fluorescence only after its proteolytic cleavage by active effector caspases, has enabled the elucidation of otherwise cryptic aspects of apoptotic regulation. In particular, we show that brief caspase activity (10 min) is sufficient to cause apoptotic death in this system. We illustrate a cytochrome c dose threshold in the oocyte, which is lowered by Smac, a protein that binds thereby neutralizing the inhibitor of apoptosis proteins. We show that meiotic oocytes develop resistance to cytochrome c, and that the eventual death of oocytes arrested in meiosis is caspase-independent. Finally, data acquired through imaging caspase activity in the Xenopus embryo suggest that apoptosis in very early development is not cell-autonomous. These studies both validate this assay as a useful tool for apoptosis research and reveal subtleties in the cell death program during early development. Moreover, this method offers a potentially valuable screening modality for identifying novel apoptotic regulators.

Of mice, birds, and men: the mouse ultrasonic song system has some features similar to humans and song-learning birds.

Humans and song-learning birds communicate acoustically using learned vocalizations. The characteristic features of this social communication behavior include vocal control by forebrain motor areas, a direct cortical projection to brainstem vocal motor neurons, and dependence on auditory feedback to develop and maintain learned vocalizations. These features have so far not been found in closely related primate and avian species that do not learn vocalizations. Male mice produce courtship ultrasonic vocalizations with acoustic features similar to songs of song-learning birds. However, it is assumed that mice lack a forebrain system for vocal modification and that their ultrasonic vocalizations are innate. Here we investigated the mouse song system and discovered that it includes a motor cortex region active during singing, that projects directly to brainstem vocal motor neurons and is necessary for keeping song more stereotyped and on pitch. We also discovered that male mice depend on auditory feedback to maintain some ultrasonic song features, and that sub-strains with differences in their songs can match each other's pitch when cross-housed under competitive social conditions. We conclude that male mice have some limited vocal modification abilities with at least some neuroanatomical features thought to be unique to humans and song-learning birds. To explain our findings, we propose a continuum hypothesis of vocal learning.

Statistical analysis of crystallization database links protein physico-chemical features with crystallization mechanisms.

X-ray crystallography is the predominant method for obtaining atomic-scale information about biological macromolecules. Despite the success of the technique, obtaining well diffracting crystals still critically limits going from protein to structure. In practice, the crystallization process proceeds through knowledge-informed empiricism. Better physico-chemical understanding remains elusive because of the large number of variables involved, hence little guidance is available to systematically identify solution conditions that promote crystallization. To help determine relationships between macromolecular properties and their crystallization propensity, we have trained statistical models on samples for 182 proteins supplied by the Northeast Structural Genomics consortium. Gaussian processes, which capture trends beyond the reach of linear statistical models, distinguish between two main physico-chemical mechanisms driving crystallization. One is characterized by low levels of side chain entropy and has been extensively reported in the literature. The other identifies specific electrostatic interactions not previously described in the crystallization context. Because evidence for two distinct mechanisms can be gleaned both from crystal contacts and from solution conditions leading to successful crystallization, the model offers future avenues for optimizing crystallization screens based on partial structural information. The availability of crystallization data coupled with structural outcomes analyzed through state-of-the-art statistical models may thus guide macromolecular crystallization toward a more rational basis.

New insights into Hoogsteen base pairs in DNA duplexes from a structure-based survey.

Hoogsteen (HG) base pairs (bps) provide an alternative pairing geometry to Watson-Crick (WC) bps and can play unique functional roles in duplex DNA. Here, we use structural features unique to HG bps (syn purine base, HG hydrogen bonds and constricted C1'-C1' distance across the bp) to search for HG bps in X-ray structures of DNA duplexes in the Protein Data Bank. The survey identifies 106 A•T and 34 G•C HG bps in DNA duplexes, many of which are undocumented in the literature. It also uncovers HG-like bps with syn purines lacking HG hydrogen bonds or constricted C1'-C1' distances that are analogous to conformations that have been proposed to populate the WC-to-HG transition pathway. The survey reveals HG preferences similar to those observed for transient HG bps in solution by nuclear magnetic resonance, including stronger preferences for A•T versus G•C bps, TA versus GG steps, and also suggests enrichment at terminal ends with a preference for 5'-purine. HG bps induce small local perturbations in neighboring bps and, surprisingly, a small but significant degree of DNA bending (∼14°) directed toward the major groove. The survey provides insights into the preferences and structural consequences of HG bps in duplex DNA.