arXiv subjects

Daniel Perry

Publications and source records attributed to Daniel Perry.

6 recordsLinked to original sources

arXiv2024

The universal Lipschitz path space of the Heisenberg group $\mathbb{H}^1$

The goal of this paper is to define and inspect a metric version of the universal path space and study its application to purely 2-unrectifiable spaces, in particular the Heisenberg group $\mathbb{H}^1$. The construction of the universal Lipschitz path space, as the metric version is called, echoes the construction of the universal cover for path-connected, locally path-connected, and semilocally simply connected spaces. We prove that the universal Lipschitz path space of a purely 2-unrectifiable space, much like the universal cover, satisfies a unique lifting property, a universal property, and is Lipschitz simply connected. The existence of such a universal Lipschitz path space of $\mathbb{H}^1$ will be used to prove that $\pi_{1}^{\text{Lip}}(\mathbb{H}^1)$ is torsion-free in a subsequent paper.

math.MG↗

arXiv2023

Existence of length minimizers in homotopy classes of Lipschitz paths in $\mathbb{H}^1$

We show that for any purely 2-unrectifiable metric space $M$, for example the Heisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carath\'{e}odory metric, every homotopy class $[\gamma]$ of Lipschitz paths contains a length minimizing representative $\gamma_\infty$ that is unique up to reparametrization. The length minimizer $\gamma_\infty$ is the core of the homotopy class $[\gamma]$ in the sense that the image of $\gamma_\infty$ is a subset of the image of any path contained in $[\gamma]$. Furthermore, the existence of length minimizers guarantees that only the trivial class in the first Lipschitz homotopy group of $M$ with a base point can be represented by a loop within each neighborhood of the base point. The results detailed here are used in arXiv:2402.10420 to define and prove properties of a universal Lipschitz path space over $\mathbb{H}^1$.

math.MG↗

arXiv2022

Compressing Cross-Lingual Multi-Task Models at Qualtrics

Experience management is an emerging business area where organizations focus on understanding the feedback of customers and employees in order to improve their end-to-end experiences. This results in a unique set of machine learning problems to help understand how people feel, discover issues they care about, and find which actions need to be taken on data that are different in content and distribution from traditional NLP domains. In this paper, we present a case study of building text analysis applications that perform multiple classification tasks efficiently in 12 languages in the nascent business area of experience management. In order to scale up modern ML methods on experience data, we leverage cross lingual and multi-task modeling techniques to consolidate our models into a single deployment to avoid overhead. We also make use of model compression and model distillation to reduce overall inference latency and hardware cost to the level acceptable for business needs while maintaining model prediction quality. Our findings show that multi-task modeling improves task performance for a subset of experience management tasks in both XLM-R and mBert architectures. Among the compressed architectures we explored, we found that MiniLM achieved the best compression/performance tradeoff. Our case study demonstrates a speedup of up to 15.61x with 2.60% average task degradation (or 3.29x speedup with 1.71% degradation) and estimated savings of 44% over using the original full-size model. These results demonstrate a successful scaling up of text classification for the challenging new area of ML for experience management.

cs.CL↗

arXiv2020

Lipschitz Homotopy Groups of Contact 3-Manifolds

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb{H}^n$ with its \cc metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger and Young (arXiv:1210.6943 [math.GT]) on the Lipschitz homotopy groups of $\mathbb{H}^1$ to an arbitrary contact 3-manifold endowed with a \cc metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(\pi,1)$-space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.

math.GT↗

arXiv2019

Controlling light emission by engineering atomic geometries in silicon photonics

By engineering atomic geometries composed of nearly 1000 atomic segments embedded in micro-resonators we observe Bragg resonances induced by the atomic lattice at the telecommunication wavelength. The geometrical arrangement of erbium atoms into a lattice inside a silicon nitride microring resonator reduces the scattering loss at a wavelength commensurate with the lattice. We confirm dependency of light emission to the atomic positions and lattice spacing and also observe Fano interference between resonant modes in the system.

physics.app-ph↗

arXiv2015

Streaming Kernel Principal Component Analysis

Kernel principal component analysis (KPCA) provides a concise set of basis vectors which capture non-linear structures within large data sets, and is a central tool in data analysis and learning. To allow for non-linear relations, typically a full $n \times n$ kernel matrix is constructed over $n$ data points, but this requires too much space and time for large values of $n$. Techniques such as the Nystr\"om method and random feature maps can help towards this goal, but they do not explicitly maintain the basis vectors in a stream and take more space than desired. We propose a new approach for streaming KPCA which maintains a small set of basis elements in a stream, requiring space only logarithmic in $n$, and also improves the dependence on the error parameter. Our technique combines together random feature maps with recent advances in matrix sketching, it has guaranteed spectral norm error bounds with respect to the original kernel matrix, and it compares favorably in practice to state-of-the-art approaches.

cs.DS↗