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Raymond Cheng

Publications and source records attributed to Raymond Cheng.

27 records · Page 2Linked to original sources
arXiv2021

Combinatorial Iterated Integrals and the Harmonic Volume of Graphs

Let $Γ$ be a connected bridgeless metric graph, and fix a point $v$ of $Γ$. We define combinatorial iterated integrals on $Γ$ along closed paths at $v$, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of $Γ$ at $v$ and the tensor algebra on the first homology of $Γ$, $\int\colon \mathbf{Z}π_1(Γ,v) \times T\mathrm{H}_1(Γ,\mathbf{R}) \to \mathbf{R}$. We show that this pairing on the two-step unipotent quotient of the group algebra allows one to recover the base-point $v$ up to well-understood finite ambiguity. We encode the data of this structure as the combinatorial harmonic volume which is valued in the tropical intermediate Jacobian. We also give a potential-theoretic characterization for hyperelliptiicity for graphs.

math.CO↗
arXiv2020

Talek: Private Group Messaging with Hidden Access Patterns

Talek is a private group messaging system that sends messages through potentially untrustworthy servers, while hiding both data content and the communication patterns among its users. Talek explores a new point in the design space of private messaging; it guarantees access sequence indistinguishability, which is among the strongest guarantees in the space, while assuming an anytrust threat model, which is only slightly weaker than the strongest threat model currently found in related work. Our results suggest that this is a pragmatic point in the design space, since it supports strong privacy and good performance: we demonstrate a 3-server Talek cluster that achieves throughput of 9,433 messages/second for 32,000 active users with 1.7-second end-to-end latency. To achieve its security goals without coordination between clients, Talek relies on information-theoretic private information retrieval. To achieve good performance and minimize server-side storage, Talek introduces new techniques and optimizations that may be of independent interest, e.g., a novel use of blocked cuckoo hashing and support for private notifications. The latter provide a private, efficient mechanism for users to learn, without polling, which logs have new messages.

cs.CR↗
arXiv2019

Ekiden: A Platform for Confidentiality-Preserving, Trustworthy, and Performant Smart Contract Execution

Smart contracts are applications that execute on blockchains. Today they manage billions of dollars in value and motivate visionary plans for pervasive blockchain deployment. While smart contracts inherit the availability and other security assurances of blockchains, however, they are impeded by blockchains' lack of confidentiality and poor performance. We present Ekiden, a system that addresses these critical gaps by combining blockchains with Trusted Execution Environments (TEEs). Ekiden leverages a novel architecture that separates consensus from execution, enabling efficient TEE-backed confidentiality-preserving smart-contracts and high scalability. Our prototype (with Tendermint as the consensus layer) achieves example performance of 600x more throughput and 400x less latency at 1000x less cost than the Ethereum mainnet. Another contribution of this paper is that we systematically identify and treat the pitfalls arising from harmonizing TEEs and blockchains. Treated separately, both TEEs and blockchains provide powerful guarantees, but hybridized, though, they engender new attacks. For example, in naive designs, privacy in TEE-backed contracts can be jeopardized by forgery of blocks, a seemingly unrelated attack vector. We believe the insights learned from Ekiden will prove to be of broad importance in hybridized TEE-blockchain systems.

cs.CR↗
arXiv2018

Inner vectors for Toeplitz operators

In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.

math.CV↗
arXiv2018

Inner functions and zero sets for $\ell^{p}_{A}$

In this paper we characterize the zero sets of functions from $\ell^{p}_{A}$ (the analytic functions on the open unit disk $D$ whose Taylor coefficients form an $\ell^p$ sequence) by developing a concept of an "inner function" modeled by Beurling's discussion of the Hilbert space $\ell^{2}_{A}$ (the classical Hardy space). The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, it is proved that when $p > 2$, there are zero sets for $\ell^{p}_{A}$ which are not Blaschke sequences.

math.CV↗
arXiv2018

Efficient Deep Learning on Multi-Source Private Data

Machine learning models benefit from large and diverse datasets. Using such datasets, however, often requires trusting a centralized data aggregator. For sensitive applications like healthcare and finance this is undesirable as it could compromise patient privacy or divulge trade secrets. Recent advances in secure and privacy-preserving computation, including trusted hardware enclaves and differential privacy, offer a way for mutually distrusting parties to efficiently train a machine learning model without revealing the training data. In this work, we introduce Myelin, a deep learning framework which combines these privacy-preservation primitives, and use it to establish a baseline level of performance for fully private machine learning.

cs.LG↗
arXiv2018

Combinatorial aspects of the quantized universal enveloping algebra of $\mathfrak{sl}_{n+1}(\mathbb{C})$

Quasi-triangular Hopf algebras were introduced by Drinfel'd in his construction of solutions to the Yang--Baxter Equation. This algebra is built upon $\mathcal{U}_h(\mathfrak{sl}_2)$, the quantized universal enveloping algebra of the Lie algebra $\mathfrak{sl}_2$. In this paper, combinatorial structure in $\mathcal{U}_h(\mathfrak{sl}_2)$ is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case $n=1$. We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel'd's $R$-matrix, but also for the arguably mysterious ribbon elements of $\mathcal{U}_h(\mathfrak{sl}_2)$. Finally, we extend these techniques to the higher dimensional algebras $\mathcal{U}_h(\mathfrak{sl}_{n+1})$. While these explicit algebraic results are well-known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.

math.CO↗
arXiv2018

Optimal Weak Parallelogram Constants for $L^p$ Spaces

Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$ \|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big), $$ $$ \|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.

math.FA↗