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Christopher Lin

Publications and source records attributed to Christopher Lin.

9 recordsLinked to original sources

Know Thy Strengths: Comprehensive Dialogue State Tracking Diagnostics

Recent works that revealed the vulnerability of dialogue state tracking (DST) models to distributional shifts have made holistic comparisons on robustness and qualitative analyses increasingly important for understanding their relative performance. We present our findings from standardized and comprehensive DST diagnoses, which have previously been sparse and uncoordinated, using our toolkit, CheckDST, a collection of robustness tests and failure mode analytics. We discover that different classes of DST models have clear strengths and weaknesses, where generation models are more promising for handling language variety while span-based classification models are more robust to unseen entities. Prompted by this discovery, we also compare checkpoints from the same model and find that the standard practice of selecting checkpoints using validation loss/accuracy is prone to overfitting and each model class has distinct patterns of failure. Lastly, we demonstrate how our diagnoses motivate a pre-finetuning procedure with non-dialogue data that offers comprehensive improvements to generation models by alleviating the impact of distributional shifts through transfer learning.

cs.CL

Torsion-free $G_2$-structures with identical Riemannian metric

Based on a general formula due to R.Bryant, we work out the topological structure of the space of torsion-free $G_2$-structures generating the same associated Riemannian metric on a compact $7$-manifold. We also identify a corresponding Lie group-theoretic structure of the space. These observations are then used to describe the moduli space of torsion-free $G_2$-structures in certain cases - by way of covering spaces.

math.DG

The Functional Determinant and the Partition Function in Geometric Flows

We propose the use of the functional determinant of geometric operators in constructing an entropy functional associated to geometric flows. Our approach is based on the direct computation of the partition function, with a well-defined set of microstates and macrostates in the canonical ensemble. The approach is motivated by a fundamental enigma in Perelman's derivation of his famous $\mathcal{W}$-entropy. The defining feature of our entropy is that the energy of each microstate in the partition function is invariant along the associated geometric flow - a clue that could be inferred from Perelman's work. Moreover, the monotonicity of our entropy along the associated geometric flow is then a natural result in the statistical mechanics framework. While we will not argue in a completely rigorous manner, we will use the formalism to derive an explicit formula for an entropy associated to conformal flows on a closed surface based on the Polyakov formula for the determinant of the Laplacian. We also discuss possible extensions of our results to more general operators and manifolds.

math.DG

The Bochner Formula via Volume Variations

In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the gradient vector of the function. Possible extensions of this approach/technique are also discussed. While the value of this approach may be limited in terms of research, we think it definitely has pedagogical value.

math.DG

Laplacian Solitons and Symmetry in G_2-geometry

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the space of symmetries (vector fields that annihilate via the Lie derivative) of a torsion-free $G_2$-structure on a compact 7-manifold is canonically isomorphic to $H^1(M,\mathbb{R})$. Some comparisons with Ricci solitons are also discussed, along with some future directions of exploration.

math.DG

Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space

In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose second fundamental form tends to zero at infinity must have bound states.

math.DG

Discrete spectrum of quantum tubes

This short paper is based on the talk of the second author given at the 2005 Hayama Symposium on Complex Analysis in Several Variables. The recent results of the authors on the spectrum of quantum tubes are summarized.

math.DG

On the Discrete Spectrum of Generalized Quantum Tubes

In this paper, we study the spectrum of quantum tubes. Under certain intrinsic assumptions of the asymptotically flat submanifold of the Euclidean space, we prove the existence of the ground state of the quantum tube. The work is a generalization of Duclos, Exner and Krejcirik (CMP, 223(1), 13-28, 2001) and ourselves(math.DG/0402252).

math.DG