abstract: In this thesis we consider the problem of facial expression recognition (FER) from video sequences. Our method is based on subspace representations and Grassmann manifold based learning. We use Local Binary Pattern (LBP) at the frame level for representing the facial features. Next we develop a model to represent the video sequence in a lower dimensional expression subspace and also as a linear dynamical system using Autoregressive Moving Average (ARMA) model. As these subspaces lie on Grassmann space, we use Grassmann manifold based learning techniques such as kernel Fisher Discriminant Analysis with Grassmann kernels for classification. We consider six expressions namely, Angry (AN), Disgust (Di), Fear (Fe), Happy (Ha), Sadness (Sa) and Surprise (Su) for classification. We perform experiments on extended Cohn-Kanade (CK+) facial expression database to evaluate the expression recognition performance. Our method demonstrates good expression recognition performance outperforming other state of the art FER algorithms. We achieve an average recognition accuracy of 97.41% using a method based on expression subspace, kernel-FDA and Support Vector Machines (SVM) classifier. By using a simpler classifier, 1-Nearest Neighbor (1-NN) along with kernel-FDA, we achieve a recognition accuracy of 97.09%. We find that to process a group of 19 frames in a video sequence, LBP feature extraction requires majority of computation time (97 %) which is about 1.662 seconds on the Intel Core i3, dual core platform. However when only 3 frames (onset, middle and peak) of a video sequence are used, the computational complexity is reduced by about 83.75 % to 260 milliseconds at the expense of drop in the recognition accuracy to 92.88 %. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2014

<div>We study the Gaudin model associated to Lie algebras of classical types.</div><div><br></div><div>First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.</div><div><br></div><div>Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian. </div>

This thesis focuses on the design of linear precoding schemes for downlink multiple-input multiple-output (MIMO) networks. These schemes are designed to be amenable to implementation in wireless networks that allow rate-limited feedback of channel state information (CSI). In the first half of this thesis, memoryless quantization codebooks are designed and incremental vector quantization techniques are developed for the representation of CSI in MIMO point-to-point links and isolated (single-cell) downlink networks. The second half of the thesis seeks to design linear precoding schemes for the multi-cell downlink networks that can achieve improved performance without requiring significantly more communication resources for CSI feedback than those required in the case of an isolated single-cell. For the quantization problem, smooth optimization algorithms are developed for the design of codebooks that possess attractive features that facilitate their implementation in practice in the addition to having good quantization properties. As one example, the proposed approach is used to design rank-2 codebooks that have a nested structure and elements from a phase-shift keying (PSK) alphabet. The designed codebooks have larger minimum distances than some existing codebooks, and provide tangible performance gains. To take advantage of temporal correlation that may exist in the wireless channel, an incremental approach to the Grassmannian quantization problem is proposed. This approach leverages existing codebooks for memoryless quantization schemes and employs a quantized form of geodesic interpolation. Two schemes that implement the principles of the proposed approach are presented. A distinguishing feature of the proposed approach is that the direction of the geodesic interpolation is specified implicitly using a point in a conventional codebook. As a result, the approach has an inherent ability to recover autonomously from errors in the feedback path. In addition to the development of the Grassmannian quantization techniques and codebooks, this thesis studies linear precoder design for the downlink MIMO networks in the cases of small networks of arbitrary topology and unbounded networks that have typical architectures. In particular, a linear precoding scheme for the isolated 2-cell network that achieves the optimal spatial degrees of freedom of the network is proposed. The implementation of a limited feedback model for the proposed linear precoding scheme is developed as well. Based on insight from that model, other linear precoding schemes that can be implemented in larger networks, but with finite size, are developed. For unbounded networks of typical architecture, such as the hexagonal arrangement of cells, linear precoding schemes that exploit the partial connectivity of the network are presented under a class of precoding schemes that is referred to as spatial reuse precoding. These precoding schemes provide substantial gains in the achievable rates of users in the network, and require only local feedback. / Thesis / Doctor of Philosophy (PhD)

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematic areas. Despite they have been studied for decades, several aspects of their geometry are still mysterious, among which identifiability and singularity of their points. In this thesis we study the secant varieties of lines of Grassmannians and of Spinor varieties. As first result, we completely determine their posets of orbits under the action of the groups SL and Spin, respectively. Then we solve the problems of identifiability and tangential-identifiability of points in the secant varieties of lines: as a consequence, we also determine the second Terracini locus to a Grassmannian and to a Spinor variety. Our main result concerns the singular locus of the secant variety of lines: we completely determine it for Grassmannians, and we give lower and upper bounds for Spinor varieties. Finally, we partially describe the poset of orbits in the secant variety of lines of any cominuscule variety.

For an affine Weyl group W, we explicitly determine the elements for which the Möbius function of the subposet of affine Grassmannians under the Bruhat order is non-zero by utilizing the quantum Bruhat graph of the classical Weyl group associated to W . Then we examine embedding stable and consistent statistics on the affine Weyl group of type A which permit the definition of an affine infinite symmetric group. / Doctor of Philosophy / Similar to the integers, there are groups that have both an infinite number of elements and also a way to partially order those elements. With a partial ordering, we can consider the interval between two elements. When we make a function that sums over an interval of elements, then we can invert the function by using something called the Mӧbius function. For many groups, the Mӧbius function is extremely unpredictable and calculating the inverse may require us to consider an infinite number of elements. In this paper, we focus on groups called affine Weyl groups, which are very useful in algebraic geometry. It turns out that most elements in these groups have a very predictable pattern in their Mӧbius functions which only considers a finite number of elements. The first part of this paper gives very simple rules for calculating it. The second part of this paper focuses on a special type of affine Weyl group: the affine symmetric groups. We provide an attempt at defining a large parent group, which we call the affine infinite symmetric group, that contains all the other affine symmetric groups.

This thesis is devoted to a thorough study of chiral rings in two-dimensional (0,2) theories. We first discuss properties of chiral operators in general two-dimensional (0,2) nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in non-twistable theories. As a special case, we study the quantum sheaf cohomology of Grassmannians as a deformation of the usual quantum cohomology. The deformation corresponds to a (0,2) deformation of the nonabelian gauged linear sigma model whose geometric phase is associated with the Grassmannian. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. Supersymmetric localization is also applicable in this case, which proves to be efficient in computing A/2 correlation functions. We then compute chiral operators in general (0,2) nonlinear sigma models, and apply them to the Gadde-Gukov-Putrov triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixed points. As another application, we extend previous works to construct (0,2) Toda-like mirrors to the sigma model engineering Grassmannians. / Ph. D.

這一篇論文是說明矩陣中的元素如果滿足熱帶格雷斯曼就可以將它用樹型圖表示。第二章簡述熱帶數學的基本運算和介紹超曲面，作為熱帶格雷斯曼的先備知識。第三章介紹演化樹，在第二節中用兩種方法重建演化樹，第一種方法UPGMA是要在特定的條件下所建造的演化樹才會正確，第二種方法是利用矩陣中兩個物種間的距離假設的樹型配置圖，解聯立方程組得到各段距離，第三節中先介紹代數幾何中的格雷斯曼再聯結到熱帶幾何中的格雷斯曼。第四章是將第二種方法寫成Python程式，輸入矩陣中的元素，如果滿足熱帶格雷斯曼，就可以顯示它可能的樹型圖以及算出距離。

In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field. This thesis concerns the construction of non-constant-dimension projective space codes for adversarial error-correction in random linear network coding. The metric used is the so-called injection distance introduced by Silva and Kschischang, which perfectly reflects the adversarial nature of the channel. A Gilbert-Varshamov-type bound for such codes is derived and its asymptotic behaviour is analysed. It is shown that in the limit as the ambient space dimension approaches infinity, the Gilbert-Varshamov bound on the size of non-constant-dimension codes behaves similar to the Gilbert-Varshamov bound on the size of constant-dimension codes contained within the largest Grassmannians in the projective space. Using the code-construction framework of Etzion and Silberstein, new non-constant-dimension codes are constructed; these codes contain more codewords than comparable codes designed for the subspace metric. To our knowledge this work is the first to address the construction of non-constant-dimension codes designed for the injection metric.

In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field. This thesis concerns the construction of non-constant-dimension projective space codes for adversarial error-correction in random linear network coding. The metric used is the so-called injection distance introduced by Silva and Kschischang, which perfectly reflects the adversarial nature of the channel. A Gilbert-Varshamov-type bound for such codes is derived and its asymptotic behaviour is analysed. It is shown that in the limit as the ambient space dimension approaches infinity, the Gilbert-Varshamov bound on the size of non-constant-dimension codes behaves similar to the Gilbert-Varshamov bound on the size of constant-dimension codes contained within the largest Grassmannians in the projective space. Using the code-construction framework of Etzion and Silberstein, new non-constant-dimension codes are constructed; these codes contain more codewords than comparable codes designed for the subspace metric. To our knowledge this work is the first to address the construction of non-constant-dimension codes designed for the injection metric.

The main topic of this thesis is based on multiple-input multiple-output (MIMO) wireless communications, which is a novel technology that has attracted great interest in the last twenty years. Conventional MIMO systems using up to eight antennas play a vital role in the urban cellular network, where the deployment of multiple antennas have significantly enhanced the throughput without taking extra spectrum or power resources. The massive MIMO systems “scales” up the benefits that offered by the conventional MIMO systems. Using sixty four or more antennas at the BS not only improves the spectrum efficiency significantly, but also provides additional link robustness. It is considered as a key technology in the fifth generation of mobile communication technology standards network, and the design of new algorithms for these two systems is the basis of the research in this thesis. Firstly, at the receiver side of the conventional MIMO systems, a general framework of bit error rate (BER) approximation for the detection algorithms is proposed, which aims to support an adaptive modulation scheme. The main idea is to utilize a simplified BER approximation scheme, which is based on the union bound of the maximum-likelihood detector (MLD), whereby the bit error rate (BER) performance of the detector for the varying channel qualities can be efficiently predicted. The K-best detector is utilized in the thesis because its quasi- MLD performance and the parallel computational structure. The simulation results have clearly shown the adaptive K-best algorithm, by applying the simplified approximation method, has much reduced computational complexity while still maintaining a promising BER performance. Secondly, in terms of the uplink channel estimation for the massive MIMO systems with the time-division-duplex operation, the performance of the Grassmannian line packing (GLP) based uplink pilot codebook design is investigated. It aims to eliminate the pilot contamination effect in order to increase the downlink achievable rate. In the case of a limited channel coherence interval, the uplink codebook design can be treated as a line packing problem in a Grassmannian manifold. The closed-form analytical expressions of downlink achievable rate for both the single-cell and multi-cell systems are proposed, which are intended for performance analysis and optimization. The numerical results validate the proposed analytical expressions and the rate gains by using the GLP-based uplink codebook design. Finally, the study is extended to the energy efficiency (EE) of the massive MIMO system, as the reduction carbon emissions from the information and communication technology is a long-term target for the researchers. An effective framework of maximizing the EE for the massive MIMO systems is proposed in this thesis. The optimization starts from the maximization of the minimum user rate, which is aiming to increase the quality-of-service and provide a feasible constraint for the EE maximization problem. Secondly, the EE problem is a non-concave problem and can not be solved directly, so the combination of fractional programming and the successive concave approximation based algorithm are proposed to find a good suboptimal solution. It has been shown that the proposed optimization algorithm provides a significant EE improvement compared to a baseline case.