Sparse differential resultant formulas: between the linear and the nonlinear case

A matrix representation of the sparse differential resultant is the basis for efficient computation algorithms, whose study promises a great contribution to the development and applicability of differential elimination techniques. It is shown how sparse linear differential resultant formulas provide bounds for the order of derivation, even in the nonlinear case, and they also provide (in many cases) the bridge with results in the nonlinear algebraic case.

New Techniques in Deep Representation Learning

Thesis (Ph.D.)--University of Washington, 2016-06

Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.

Clustering web documents using hierarchical representation with multi-granularity

Web document cluster analysis plays an important role in information retrieval by organizing large amounts of documents into a small number of meaningful clusters. Traditional web document clustering is based on the Vector Space Model (VSM), which takes into account only two-level (document and term) knowledge granularity but ignores the bridging paragraph granularity. However, this two-level granularity may lead to unsatisfactory clustering results with “false correlation”. In order to deal with the problem, a Hierarchical Representation Model with Multi-granularity (HRMM), which consists of five-layer representation of data and a twophase clustering process is proposed based on granular computing and article structure theory. To deal with the zero-valued similarity problemresulted from the sparse term-paragraphmatrix, an ontology based strategy and a tolerance-rough-set based strategy are introduced into HRMM. By using granular computing, structural knowledge hidden in documents can be more efficiently and effectively captured in HRMM and thus web document clusters with higher quality can be generated. Extensive experiments show that HRMM, HRMM with tolerancerough-set strategy, and HRMM with ontology all outperform VSM and a representative non VSM-based algorithm, WFP, significantly in terms of the F-Score.

Noradrenergic modulatoins of odor learning and odor representation in the rat

How experience alters neuronal ensemble dynamics and how locus coeruleus-mediated norepinephrine release facilitates memory formation in the brain are the topics of this thesis. Here we employed a visualization technique, cellular compartment analysis of temporal activity by fluorescence in situ hybridization (catFISH), to assess activation patterns of neuronal ensembles in the olfactory bulb (OB) and anterior piriform cortex (aPC) to repeated odor inputs. Two associative learning models were used, early odor preference learning in rat pups and adult rat go-no-go odor discrimination learning. With catFISH of an immediate early gene, Arc, we showed that odor representation in the OB and aPC was sparse (~5-10%) and widely distributed. Odor associative learning enhanced the stability of the rewarded odor representation in the OB and aPC. The stable component, indexed by the overlap between the two ensembles activated by the rewarded odor at two time points, increased from ~25% to ~50% (p = 0.004-1.43E⁻4; Chapter 3 and 4). Adult odor discrimination learning promoted pattern separation between rewarded and unrewarded odor representations in the aPC. The overlap between rewarded and unrewarded odor representations reduced from ~25% to ~14% (p = 2.28E⁻⁵). However, learning an odor mixture as a rewarded odor increased the overlap of the component odor representations in the aPC from ~23% to ~44% (p = 0.010; Chapter 4). Blocking both α- and β-adrenoreceptors in the aPC prevented highly similar odor discrimination learning in adult rats, and reduced OB mitral and granule ensemble stability to the rewarded odor. Similar treatment in the OB only slowed odor discrimination learning. However, OB adrenoceptor blockade disrupted pattern separation and ensemble stability in the aPC when the rats demonstrated deficiency in discrimination (Chapter 5). In another project, the role of α₂-adrenoreceptors in the OB during early odor preference learning was studied. OB α2-adrenoceptor activation was necessary for odor learning in rat pups. α₂-adrenoceptor activation was additive with β-adrenoceptor mediated signalling to promote learning (Chapter 2). Together, these experiments suggest that odor representations are highly adaptive at the early stages of odor processing. The OB and aPC work in concert to support odor learning and top-down adrenergic input exerts a powerful modulation on both learning and odor representation.

Sparse time-frequency data analysis : a multi-scale approach

In this work, we further extend the recently developed adaptive data analysis method, the Sparse Time-Frequency Representation (STFR) method. This method is based on the assumption that many physical signals inherently contain AM-FM representations. We propose a sparse optimization method to extract the AM-FM representations of such signals. We prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic STFR, which extends the method to tackle problems that former STFR methods could not handle, including stability to noise and non-periodic data analysis. This is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. Moreover, we propose a new STFR algorithm to study intrawave signals with strong frequency modulation and analyze the convergence of this new algorithm for periodic signals. Such signals have previously remained a bottleneck for all signal processing methods. Furthermore, we propose a modified version of STFR that facilitates the extraction of intrawaves that have overlaping frequency content. We show that the STFR methods can be applied to the realm of dynamical systems and cardiovascular signals. In particular, we present a simplified and modified version of the STFR algorithm that is potentially useful for the diagnosis of some cardiovascular diseases. We further explain some preliminary work on the nature of Intrinsic Mode Functions (IMFs) and how they can have different representations in different phase coordinates. This analysis shows that the uncertainty principle is fundamental to all oscillating signals.