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Gang Liao

Publications and source records attributed to Gang Liao.

At least 19 recordsLinked to original sources
arXiv2026

A prime orbit theorem for smooth surface diffeomorphisms

We establish a sharp prime orbit theorem for every homoclinic class of a $C^\infty$ diffeomorphism on a closed surface with positive topological entropy. Let $\mathcal{H}$ be a homoclinic class with topological entropy $h > 0$. Then there exists a constant $\chi_2 < 0$ such that for any $\chi_1 \in (0, h)$, \[ \lim_{\substack{l(\mathcal{H}) \mid n \\ n\to\infty}} \frac{\sharp P_{\chi_1,\chi_2}(n)}{e^{nh}} = l(\mathcal{H}). \] Here $P_{\chi_1,\chi_2}(n)$ stands for the set of period-$n$ saddle points in $\mathcal{H}$ with Lyapunov exponents lying outside the interval $[\chi_2,\chi_1]$, and $l(\mathcal{H})$ denotes the period associated with the homoclinic class $\mathcal{H}$.

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arXiv2026

Experience Graphs: The Data Foundation for Self-Improving Agents

The database community has repeatedly advanced the state of the art by recognizing that new workloads demand new system architectures. We argue that long-horizon agentic tasks -- code generation, scientific discovery, hardware design -- are such a workload. These agents explore: they generate artifacts, execute tools, observe failures, branch, and repair over hundreds of steps. This search produces a structured object we call an experience graph: executable artifacts, tool outputs, rewards, sibling comparisons, and causal lineage. Yet existing agent frameworks treat this experience as disposable state -- JSON checkpoints and session logs that cannot be recovered after a crash, queried across users, or materialized into training data. We propose Trellis: a data foundation that treats the experience graph as first-class, governed, queryable database state. The core insight is that search over experience graphs is a database access pattern. Frontier selection is a query, cross-session reuse is vector-seeded graph retrieval, training-data extraction is a materialized view, and reconstructing what an agent knew at any past step is a time-travel query. When the database owns the experience graph, agents become stateless compute, and crash recovery, horizontal scaling, and a closed-loop training flywheel emerge as architectural byproducts. We ground the design in KernelEvolve, a production accelerator-kernel optimizer at Meta, where cross-session reuse reaches a target speedup roughly 10x faster at 52% lower token cost. More broadly, Trellis turns inference-time search from disposable computation into a durable institutional asset: logs made databases reliable; experience graphs may make agents cumulative.

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arXiv2026

Continuity properties of partial entropy

We establish a general criterion on the upper semi-continuity of partial entropy in all directions for $C^{1+\alpha}$ diffeomorphisms: it holds when the respective sums of Lyapunov exponents are continuous. This addresses, in arbitrary dimensions, the converse aspect of the entropic continuity of the Lyapunov exponents established by Buzzi, Crovisier, and Sarig. Consequently, the entropy (and all the partial entropies) is always upper semi-continuous at generic ergodic measures of every $C^{1+\alpha}$ diffeomorphism, which extends the $C^{\infty}$ result of Newhouse. Numerous applications and examples are provided, including topics related to measures with dominated splittings, SRB measures, average expanding diffeomorphisms, singular flows, standard maps, and symbolic codings for diffeomorphisms.

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arXiv2025

KernelEvolve: Scaling Agentic Kernel Coding for Heterogeneous AI Accelerators at Meta

Making deep learning recommendation model (DLRM) training and inference fast and efficient is important. However, this presents three key system challenges - model architecture diversity, kernel primitive diversity, and hardware generation and architecture heterogeneity. This paper presents KernelEvolve-an agentic kernel coding framework-to tackle heterogeneity at-scale for DLRM. KernelEvolve is designed to take kernel specifications as input and automate the process of kernel generation and optimization for recommendation model across heterogeneous hardware architectures. KernelEvolve does so by operating at multiple programming abstractions, from Triton and CuTe DSL to low-level hardware agnostic languages, spanning the full hardware-software optimization stack. The kernel optimization process is described as graph-based search with selection policy, universal operator, fitness function, and termination rule, dynamically adapts to runtime execution context through retrieval-augmented prompt synthesis. We designed, implemented, and deployed KernelEvolve to optimize a wide variety of production recommendation models across generations of NVIDIA and AMD GPUs, as well as Meta's AI accelerators. We validate KernelEvolve on the publicly-available KernelBench suite, achieving 100% pass rate on all 250 problems across three difficulty levels, and 160 PyTorch ATen operators across three heterogeneous hardware platforms, demonstrating 100% correctness. KernelEvolve reduces development time from weeks to hours and achieves substantial performance improvements over PyTorch baselines across diverse production use cases and for heterogeneous AI systems at-scale. Beyond performance efficiency improvements, KernelEvolve significantly mitigates the programmability barrier for new AI hardware by enabling automated kernel generation for in-house developed AI hardware.

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arXiv2025

Metric entropy and homoclinic growth rate

In this paper, we investigate the relationship between chaos and homoclinic orbits from a quantitative perspective. Let f be a C^r diffeomorphism (r > 1) on a compact Riemannian manifold preserving an ergodic hyperbolic measure. We show that the homoclinic growth rate is bounded below by the metric entropy. This result generalizes the work of Mendoza from surfaces to higher-dimensional systems from a measure-theoretic viewpoint. We also examine the sharpness of this estimate by demonstrating that, in the Newhouse domain, C^r-generic diffeomorphisms exhibit a superexponential growth in the number of homoclinic points.

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arXiv2025

Quasi-shadowing property for nonuniformly partially hyperbolic systems

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a $C^{1+\alpha}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property and quasi-closing property are also investigated. As an application of quasi-closing property, we extend Katok's reslut on the growth of periodoc orbits for hyperbolic ergodic measure to any ergodic measure: the number of quasi-periodic points grows exponentially at least the metric entropy.

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arXiv2024

Entropy formula of folding type for $C^{1+\alpha}$ maps

In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} \[h_{\mu}(f)= F_{\mu}(f)-\displaystyle\int\sum\nolimits_{\lambda_i(x)<0} \lambda_i(x)d\mu(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+\alpha}$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $\mu$ is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general $C^{1+\alpha}$ maps.

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arXiv2024

Flock: A Low-Cost Streaming Query Engine on FaaS Platforms

Existing serverless data analytics systems rely on external storage services like S3 for data shuffling and communication between cloud functions. While this approach provides the elasticity benefits of serverless computing, it incurs additional latency and cost overheads. We present Flock, a novel cloud-native streaming query engine that leverages the on-demand scalability of FaaS platforms for real-time data analytics. Flock utilizes function invocation payloads for efficient data exchange, eliminating the need for external storage. This not only reduces latency and cost but also simplifies the architecture by removing the requirement for a centralized coordinator. Flock employs a template-based approach to dynamically create cloud functions for each query stage and a function group mechanism for handling data aggregation and shuffling. It supports both SQL and DataFrame APIs, making it easy to use. Our evaluation shows that Flock provides significant performance gains and cost savings compared to existing serverless and serverful streaming systems. It outperforms Apache Flink by 10-20x in cost while achieving similar latency and throughput.

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arXiv2024

Bullion: A Column Store for Machine Learning

The past two decades have witnessed significant success in applying columnar storage to data warehousing and analytics. However, the rapid growth of machine learning poses new challenges. This paper presents Bullion, a columnar storage system tailored for machine learning workloads. Bullion addresses the complexities of data compliance, optimizes the encoding of long sequence sparse features, efficiently manages wide-table projections, introduces feature quantization in storage, enables quality-aware sequential reads for multimodal training data, and provides a comprehensive cascading encoding framework that unifies diverse encoding schemes through modular, composable interfaces. By aligning with the evolving requirements of ML applications, Bullion facilitates the application of columnar storage and processing to modern application scenarios such as those within advertising, recommendation systems, and Generative AI. Preliminary experimental results and theoretical analysis demonstrate Bullion's improved ability to deliver strong performance in the face of the unique demands of machine learning workloads compared to existing columnar storage solutions. Bullion significantly reduces I/O costs for deletion compliance, achieves substantial storage savings with its optimized encoding scheme for sparse features, and improves metadata parsing speed for wide-table projections. These advancements enable Bullion to become an important component in the future of machine learning infrastructure, enabling organizations to efficiently manage and process the massive volumes of data required for training and inference in modern AI applications.

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arXiv2024

SFVInt: Simple, Fast and Generic Variable-Length Integer Decoding using Bit Manipulation Instructions

The ubiquity of variable-length integers in data storage and communication necessitates efficient decoding techniques. In this paper, we present SFVInt, a simple and fast approach to decode the prevalent Little Endian Base-128 (LEB128) varints. Our approach effectively utilizes the Bit Manipulation Instruction Set 2 (BMI2) in modern Intel and AMD processors, achieving significant performance improvement while maintaining simplicity and avoiding overengineering. SFVInt, with its generic design, effectively processes both 32-bit and 64-bit unsigned integers using a unified code template, marking a significant leap forward in varint decoding efficiency. We thoroughly evaluate SFVInt's performance across various datasets and scenarios, demonstrating that it achieves up to a 2x increase in decoding speed when compared to varint decoding methods used in established frameworks like Facebook Folly and Google Protobuf.

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arXiv2020

Margulis-Ruelle inequality for general manifolds

In this paper we investigate the Margulis-Ruelle inequality for general Riemannian manifolds (possibly noncompact and with boundary) and show that it always holds under integrable condition.

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arXiv2020

Uniform tail entropy for real analytic maps

Let $M$ be a compact real analytic manifold of finite dimension. There is a function $a: (0,+\infty)\to [0,+\infty)$ with $\lim_{t\to0}a(t)=0$ such that, the tail entropy $h^{*}(f,\varepsilon)$ of any real analytic map $f$ on $M$ is uniformly bounded above by the scale $a(\varepsilon)$.

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arXiv2020

Continuity properties of folding entropy

The folding entropy is a quantity originally proposed by Ruelle in 1996 during the study of entropy production in the non-equilibrium statistical mechanics. As derived through a limiting process to the non-equilibrium steady state, the continuity of entropy production plays a key role in its physical interpretations. In this paper, we study the continuity of folding entropy for a general (non-invertible) differentiable dynamical system with degeneracy. By introducing a notion called degenerate rate, we prove that on any subset of measures with uniform degenerate rate, the folding entropy, and hence the entropy production, is upper semi-continuous. This extends the upper semi-continuity result from endomorphisms to all $C^r(r>1)$ maps. We further apply in the one-dimensional setting. In achieving this, an equality between the folding entropy and (Kolmogorov-Sinai) metric entropy, as well as a general dimension formula are established. These admit their own interests. The upper semi-continuity of metric entropy and dimension are then valid when measures with uniform degenerate rate are considered. Moreover, the sharpness of uniform degenerate rate is also investigated by examples in the scope of positive metric (or folding) entropy.

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arXiv2019

Symbolic extensions for 3-dimensional diffeomorphisms

We prove that every $\mathcal{C}^{r}$ diffeomorphism with $r>1$ on a three-dimensional manifold admits symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. This answers positively a conjecture of Downarowicz and Newhouse in dimension three.

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arXiv2016

Variational equalities of entropy in nonuniformly hyperbolic systems

In this paper we prove that for an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $ω$-full measured set $\widetildeΛ$ such that for every invariant probability $μ\in \mathcal{M}_{inv}(\widetildeΛ,f)$, the metric entropy of $μ$ is equal to the topological entropy of saturated set $G_μ$ consisting of generic points of $μ$: $$h_μ(f)=h_{\top}(f,G_μ).$$ Moreover, for every nonempty, compact and connected subset $K$ of $\mathcal{M}_{inv}(\widetildeΛ,f)$ with the same hyperbolic rate, we compute the topological entropy of saturated set $G_K$ of $K$ by the following equality: $$\inf\{h_μ(f)\mid μ\in K\}=h_{\top}(f,G_K).$$ In particular these results can be applied (i) to the nonuniformy hyperbolic diffeomorphisms described by Katok, (ii) to the robustly transitive partially hyperbolic diffeomorphisms described by ~Ma{ñ}{é}, (iii) to the robustly transitive non-partially hyperbolic diffeomorphisms described by Bonatti-Viana. In all these cases $\mathcal{M}_{inv}(\widetildeΛ,f)$ contains an open subset of $\mathcal{M}_{erg}(M,f)$.

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arXiv2015

Ultra-fast Multiple Genome Sequence Matching Using GPU

In this paper, a contrastive evaluation of massively parallel implementations of suffix tree and suffix array to accelerate genome sequence matching are proposed based on Intel Core i7 3770K quad-core and NVIDIA GeForce GTX680 GPU. Besides suffix array only held approximately 20%~30% of the space relative to suffix tree, the coalesced binary search and tile optimization make suffix array clearly outperform suffix tree using GPU. Consequently, the experimental results show that multiple genome sequence matching based on suffix array is more than 99 times speedup than that of CPU serial implementation. There is no doubt that massively parallel matching algorithm based on suffix array is an efficient approach to high-performance bioinformatics applications.

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arXiv2014

Continuity of entropy map for nonuniformly hyperbolic systems

We prove that entropy map is upper semi-continuous for C1 nonuniformly hyperbolic systems with domination, while it is not true for C1+alpha nonuniformly hyperbolic systems in general. This goes a little against a common intuition that conclusions are parallel between C1+domination systems and C1+alpha systems.

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