arXiv subjects

Yi Lin

Publications and source records attributed to Yi Lin.

At least 145 records · Page 8Linked to original sources

arXiv2019

Kirwan Surjectivity for the Equivariant Dolbeault cohomology

Consider the holomorphic Hamiltonian action of a compact Lie group $K$ on a compact Kähler manifold $M$ with a moment map $Φ: M\rightarrow \mathfrak{k}^*$. Assume that $0$ is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient $M_0:=Φ^{-1}(0)/K$ if all the ordinary cohomology of $M$ is of type $(p, p)$. In this paper, using the Cartan-Chern-Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of $M$ onto the Dolbeault cohomology of the Kähler quotient $M_0$. As an immediate consequence, this result provides an answer to the question posed by Weitsman.

math.SG↗

arXiv2019

On optimal designs for non-regular models

Classically, Fisher information is the relevant object in defining optimal experimental designs. However, for models that lack certain regularity, the Fisher information does not exist and, hence, there is no notion of design optimality available in the literature. This article seeks to fill the gap by proposing a so-called Hellinger information, which generalizes Fisher information in the sense that the two measures agree in regular problems, but the former also exists for certain types of non-regular problems. We derive a Hellinger information inequality, showing that Hellinger information defines a lower bound on the local minimax risk of estimators. This provides a connection between features of the underlying model---in particular, the design---and the performance of estimators, motivating the use of this new Hellinger information for non-regular optimal design problems. Hellinger optimal designs are derived for several non-regular regression problems, with numerical results empirically demonstrating the efficiency of these designs compared to alternatives.

math.ST↗

arXiv2019

DBSCAN for nonlinear equalization in high-capacity multi-carrier optical communications

Coherent optical multi-carrier communications have recently dominated metro-regional and long-haul optical communications. However, the major obstacle of networks involving coherent multi-carrier signals such as coherent optical orthogonal frequency-division multiplexing (CO-OFDM) is the fiber-induced nonlinearity and the parametric noise amplification from cascaded optical amplifiers which results in significant nonlinear distortion among subcarriers. Here, we present the first nonlinear equalizer in optical communications using the traditional Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm and a novel modified version of DBSCAN which combines K-means clustering on the noisy un-clustered symbols. For a 24.72 Gbit/sec differential quaternary phase-shift keying (DQPSK) CO-OFDM system, the modified DBSCAN can increase the signal quality-factor by up to 2.158 dB compared to linear equalization at 500 km of transmission. The modified DBSCAN slightly outperforms the traditional DBSCAN, fuzzy-logic C-means, hierarchical and conventional K-means clustering at high launched optical powers.

eess.SP↗

arXiv2018

Semi-supervised mp-MRI Data Synthesis with StitchLayer and Auxiliary Distance Maximization

In this paper, we address the problem of synthesizing multi-parameter magnetic resonance imaging (mp-MRI) data, i.e. Apparent Diffusion Coefficients (ADC) and T2-weighted (T2w), containing clinically significant (CS) prostate cancer (PCa) via semi-supervised adversarial learning. Specifically, our synthesizer generates mp-MRI data in a sequential manner: first generating ADC maps from 128-d latent vectors, followed by translating them to the T2w images. The synthesizer is trained in a semisupervised manner. In the supervised training process, a limited amount of paired ADC-T2w images and the corresponding ADC encodings are provided and the synthesizer learns the paired relationship by explicitly minimizing the reconstruction losses between synthetic and real images. To avoid overfitting limited ADC encodings, an unlimited amount of random latent vectors and unpaired ADC-T2w Images are utilized in the unsupervised training process for learning the marginal image distributions of real images. To improve the robustness of synthesizing, we decompose the difficult task of generating full-size images into several simpler tasks which generate sub-images only. A StitchLayer is then employed to fuse sub-images together in an interlaced manner into a full-size image. To enforce the synthetic images to indeed contain distinguishable CS PCa lesions, we propose to also maximize an auxiliary distance of Jensen-Shannon divergence (JSD) between CS and nonCS images. Experimental results show that our method can effectively synthesize a large variety of mpMRI images which contain meaningful CS PCa lesions, display a good visual quality and have the correct paired relationship. Compared to the state-of-the-art synthesis methods, our method achieves a significant improvement in terms of both visual and quantitative evaluation metrics.

cs.LG↗

arXiv2018

Equivariant formality of transversely symplectic foliations and Frobenius manifolds

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation which satisfies the transverse hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic $dδ$-lemma in this setting. As an application, we show that if the foliation is also Riemannian, then there exists a natural formal Frobenius manifold structure on the equivariant basic cohomology of the foliation.

math.SG↗

arXiv2017

Convexity properties of presymplectic moment maps

The convexity and Morse-theoretic properties of moment maps in symplectic geometry typically fail for presymplectic manifolds. We find a condition on presymplectic moment maps that prevents these failures. Our result applies for instance to Prato's quasifolds and to Hamiltonian actions on contact manifolds and cosymplectic manifolds.

math.SG↗

arXiv2016

Hodge theory on transversely symplectic foliations

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic $dδ$-lemma for any such foliations with the (transverse) $s$-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact $K$-contact manifolds, the $s$-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a $2n+1$ dimensional compact $K$-contact manifold with the (transverse) $s$-Lefschetz property is at most $2n-s$. For any even integer $s\geq 2$, we also apply our main result to produce examples of $K$-contact manifolds that are $s$-Lefschetz but not $(s+1)$-Lefschetz.

math.SG↗

arXiv2016

Lefschetz contact manifolds and odd dimensional symplectic geometry

In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology of a compact Sasakian manifold satisfies the transverse Lefschetz property. The second version, established far more recently by Cappelletti-Montano, De Nicola, and Yudin, holds for the De Rham cohomology of a compact Sasakian manifold. In the current paper, using the formalism of odd dimensional symplectic geometry, we prove a Hard Lefschetz theorem for a compact $K$-contact manifold, which implies immediately that the two existing versions of Hard Lefschetz theorems are mathematically equivalent to each other. Our method sheds new light on the Hard Lefschetz property of a Sasakian manifold. It enables us to give a simple construction of simply-connected $K$-contact manifolds without any Sasakian structures in any dimension $\geq 9$, and answers an open question asked by Boyer and late Galicki concerning the existence of such examples. It also allows us to establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a compact Lefschetz contact five manifold. As an immediate application, we use it to produce first examples of compact Lefschetz contact manifolds which do not support any Sasakian structures. This provides an answer to another open question asked by Cappelletti-Montano, De Nicola, and Yudin in their recent work.

math.SG↗

arXiv2016

Exact prior-free probabilistic inference in a class of non-regular models

The use of standard statistical methods, such as maximum likelihood, is often justified based on their asymptotic properties. For suitably regular models, this theory is standard but, when the model is non-regular, e.g., the support depends on the parameter, these asymptotic properties may be difficult to assess. Recently, an inferential model (IM) framework has been developed that provides valid prior-free probabilistic inference without the need for asymptotic justification. In this paper, we construct an IM for a class of highly non-regular models with parameter-dependent support. This construction requires conditioning, which is facilitated through the solution of a particular differential equation. We prove that the plausibility intervals derived from this IM are exact confidence intervals, and we demonstrate their efficiency in a simulation study.

stat.ME↗

arXiv2013

Symplectic Harmonic theory and the Federer-Fleming deformation theorem

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fundamental property on symplectic Harmonic forms. We show that on a closed symplectic manifold, every real primitive cohomology class of positive degrees admits a symplectic Harmonic representative not supported on the entire mani- fold. As an application, we use it to investigate the support of symplectic Harmonic representatives of Thom classes, and give a complete solution to an open question asked by Guillemin.

math.SG↗

arXiv2012

Log-concavity and symplectic flows

Let M be a compact, connected symplectic 2n-dimensional manifold on which an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic orbits, we show that the Duistermaat-Heckman measure of the T-action is log-concave. This verifies the logarithmic concavity conjecture for a class of inequivalent T-actions. Then we use this conjecture to prove the following: if there is an effective symplectic action of an (n-2)-dimensional torus T on a compact, connected symplectic 2n-dimensional manifold that admits an effective complementary symplectic action of a 2-torus with symplectic orbits, then the existence of T-fixed points implies that the T-action is Hamiltonian. As a consequence of this, we give new proofs of a classical theorem by McDuff about S^1-actions, and some of its recent extensions.

math.SG↗

arXiv2009

Generalized complex hamiltonian torus actions: Examples and constraints

Consider an effective Hamiltonian torus action $T\times M \to M$ on a topologically twisted,generalized complex manifold $M$ of dimension $2n$. We prove that the $rank(T) \leq n-2$ and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying $rank(T) = n-2$, using a surgery procedure on toric manifolds.

math.DG↗

arXiv2008

Topology of generalized complex quotients

Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold $M$. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.

math.DG↗

arXiv2008

The log-concavity conjecture for the Duistermaat-Heckman measure revisited

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed points set is the disjoint union of two copies of $T^4$. In this article, for any closed symplectic four manifold $N$ with $b+$ greater than 1, we show that there is a Hamiltonian six manifold $M$ such that its fixed points set is the disjoint union of two copies of $N$ and such that its Duistermaat-Heckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity two such that all the symplectic reduced spaces taken at regular values satisfy the condition $b+=1$, then its Duistermaat-Heckman function has to be log-concave. As a consequence, we prove the log-concavity conjecture for Hamiltonian circle actions on six manifolds such that the fixed points sets have no four dimensional components, or only have four dimensional pieces with $b+=1$.

math.SG↗

arXiv2007

The Equivariant cohomology theory of twisted generalized complex manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general $(2,2)$ gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on $H$-twisted generalized complex manifolds. If the manifold satisfies the $\bar{\partial}\partial$-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kähler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an $H$-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure. As a side result, we show in this paper that the generalized Kähler quotient of a generalized Kähler vector space can never have a (cohomologically) non-trivial twisting. This gives a negative answer to a question asked by physicists whether one can construct $(2,2)$ gauged linear sigma models with non-trivial fluxes.

math.DG↗

arXiv2007

Generalized geometry, equivariant $\bar{\partial}\partial$-lemma, and torus actions

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold $M$ satisfies the $\bar{\partial}\partial$-lemma, we prove that they are both canonically isomorphic to $(S\g^*)^G\otimes H_H(M)$, where $(S\g^*)^G$ is the space of invariant polynomials over the Lie algebra $\g$ of $G$, and $H_H(M)$ is the $H$-twisted cohomology of $M$. Furthermore, we establish an equivariant version of the $\bar{\partial}\partial$-lemma, namely $\bar{\partial}_G\partial$-lemma, which is a direct generalization of the $d_Gδ$-lemma for Hamiltonian symplectic manifolds with the Hard Lefschetz property. Second we consider the torus action on a compact generalized Kähler manifold which preserves the generalized Kähler structure and which is equivariantly formal. We prove a generalization of a result of Carrell and Lieberman in generalized Kähler geometry. We then use it to compute the generalized Hodge numbers for non-trivial examples of generalized Kähler structures on $\C¶^n$ and $\CP^n$ blown up at a fixed point.

math.DG↗

arXiv2007

Component selection and smoothing in multivariate nonparametric regression

We propose a new method for model selection and model fitting in multivariate nonparametric regression models, in the framework of smoothing spline ANOVA. The ``COSSO'' is a method of regularization with the penalty functional being the sum of component norms, instead of the squared norm employed in the traditional smoothing spline method. The COSSO provides a unified framework for several recent proposals for model selection in linear models and smoothing spline ANOVA models. Theoretical properties, such as the existence and the rate of convergence of the COSSO estimator, are studied. In the special case of a tensor product design with periodic functions, a detailed analysis reveals that the COSSO does model selection by applying a novel soft thresholding type operation to the function components. We give an equivalent formulation of the COSSO estimator which leads naturally to an iterative algorithm. We compare the COSSO with MARS, a popular method that builds functional ANOVA models, in simulations and real examples. The COSSO method can be extended to classification problems and we compare its performance with those of a number of machine learning algorithms on real datasets. The COSSO gives very competitive performance in these studies.

math.ST↗

arXiv2006

Examples of Non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property

In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counter examples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong Lefschetz manifold. We then use it to show the existence of Lefschetz four-manifolds with non-Lefschetz finite covering spaces.

math.SG↗