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Tiancheng Xu

Publications and source records attributed to Tiancheng Xu.

8 recordsLinked to original sources
arXiv2026

Learning to Live: A Step Toward Lifelong Embodied Intelligence

Embodied agents are usually organized around individual tasks, yet the physical world keeps changing between tasks and during execution. We introduce LIFE (Lifelong Intelligence from Experience), a framework in which a robot observes continuously, remembers selectively, and turns experience into new capabilities. RoboBeats, a family of lightweight streaming gates, tracks conditions set by the agent and discovers new events, deciding when an observation warrants a response or a memory write. Selected observations enter Plastic Memory, a flat store of images and structured records that keeps the source evidence, so new fields and indexes can be added when later questions need them. When existing tools fall short, Explore-Distill-Mount (EDM) turns solutions found during problem solving into validated, reusable tools and skills. Under matched protocols, RoboBeats substantially improves condition tracking over its base models and, at the same writing budget, retains more useful evidence for memory question answering than uniform sampling. A case study of limited scale, replaying 24 hours of continuous video recorded by a real robot, illustrates how the integrated system builds memory from a day of experience and uses it to answer everyday questions. LIFE takes a step toward robots that become more familiar with their surroundings and more capable through experience: learning not only to complete tasks, but to live in the world.

cs.RO↗
arXiv2026

Optimal Configurations for Mordell's Problem

Mordell's problem asks for the largest product of squared pairwise distances among $n$ points in the complex plane with a prescribed sum of squared distances from the origin. When this sum equals $n$, the maximum for $2\le n\le5$ is known to be $n^n$, attained precisely by the $n$th roots of unity up to rotation and relabeling. For $n\ge6$, Chalk showed that a regular polygon together with its center gives a larger product than the regular $n$-gon. We prove that for $6\le n\le9$ the maximum is $(n-1)^{n-1}r_n^{n(n-1)}$, where $r_n=\sqrt{n/(n-1)}$. Up to the same symmetries, the unique maximizer consists of the origin and the vertices of a regular $(n-1)$-gon of radius $r_n$. For six points, algebraic elimination reduces the proof to an exact comparison of stationary values. For seven points, we compare all twelve classes in a published classification of equilibria. For eight and nine points, rigorous interval arithmetic establishes an inequality for every admissible pair. Summing over pairs gives the sharp global bound, and the equality conditions determine the configuration. An explicit ten-point comparison and a known instability result together exclude Chalk's configuration for every $n\ge10$. For each $n$ from two through eight, Mordell's problem and the minimum-diameter problem studied by Bateman and Erdős have a common optimal configuration.

math.CO↗
arXiv2026

On the Five-Point Case of Mordell's Problem

For five complex points $z_1,\ldots,z_5$ satisfying $\sum_i|z_i|^2=5$, Mordell's discriminant inequality states that $\prod_{i<j}|z_i-z_j|^2\leq5^5$, with equality precisely for a unit regular pentagon, up to rotation and permutation. We organize its proof around a direct comparison of discriminant values. A global maximizer gives a centered stationary configuration, whose coefficient equations reduce the generic case to two quartic branches. On each branch, the absolute value of a cubic polynomial gives the original discriminant objective after accounting for the normalization scale. Exact rational interval bounds exclude both branches from the maximum, and the exceptional coefficient cases leave the regular pentagon. The argument determines the maximizing configuration without reconstructing all stationary shapes. We have completed a full Lean formalization of the five-point inequality and both directions of its equality condition, including all reductions and finite checks used in this proof.

math.CV↗
arXiv2026

A Finite Certificate for the Positive $n=11$ Vasc Inequality

We establish the Vasc cyclic inequality for all strictly positive real eleven-tuples by a finite exact certificate. Cyclic rotation places a minimum at the first position, and rank words together with cumulative gaps reduce the problem to $10!=3{,}628{,}800$ homogeneous integer polynomials of degree eleven. Independent coefficient reconstruction proves $3{,}358{,}617$ roots directly. The remaining $270{,}183$ roots are covered by $267{,}952$ ordinary midpoint certificates, $2{,}195$ complete binary subdivision certificates, and $36$ complete certificates containing partial sorting nodes. Each nontrivial leaf is checked by subtracting weighted AM-GM midpoint circuits from a positive multiple of the transformed polynomial and verifying every residual coefficient. Explicit word-set comparisons certify disjointness and exact coverage. We prove the soundness of the leaves and both subdivision rules, then recover the rational inequality by positive denominator clearing and continuity. The resulting proof combines explicit mathematical reductions with independently checked exact integer certificates.

cs.SC↗
arXiv2025

Design and accuracy trade-offs in Computational Statistics

Statistical computations are becoming increasingly important. These computations often need to be performed in log-space because probabilities become extremely small due to repeated multiplications. While using logarithms effectively prevents numerical underflow, this paper shows that its cost is high in performance, resource utilization, and, notably, numerical accuracy. This paper then argues that using posit, a recently proposed floating-point format, is a better strategy for statistical computations operating on extremely small numbers because of its unique encoding mechanism. To that end, this paper performs a comprehensive analysis comparing posit, binary64, and logarithm representations, examining both individual arithmetic operations, statistical bioinformatics applications, and their accelerators. FPGA implementation results highlight that posit-based accelerators can achieve up to two orders of magnitude higher accuracy, up to 60\% lower resource utilization, and up to $1.3\times$ speedup, compared to log-space accelerators. Such improvement translates to $2\times$ performance per unit resource on the FPGA.

math.NA↗
arXiv2020

Mesorasi: Architecture Support for Point Cloud Analytics via Delayed-Aggregation

Point cloud analytics is poised to become a key workload on battery-powered embedded and mobile platforms in a wide range of emerging application domains, such as autonomous driving, robotics, and augmented reality, where efficiency is paramount. This paper proposes Mesorasi, an algorithm-architecture co-designed system that simultaneously improves the performance and energy efficiency of point cloud analytics while retaining its accuracy. Our extensive characterizations of state-of-the-art point cloud algorithms show that, while structurally reminiscent of convolutional neural networks (CNNs), point cloud algorithms exhibit inherent compute and memory inefficiencies due to the unique characteristics of point cloud data. We propose delayed-aggregation, a new algorithmic primitive for building efficient point cloud algorithms. Delayed-aggregation hides the performance bottlenecks and reduces the compute and memory redundancies by exploiting the approximately distributive property of key operations in point cloud algorithms. Delayed-aggregation let point cloud algorithms achieve 1.6x speedup and 51.1% energy reduction on a mobile GPU while retaining the accuracy (-0.9% loss to 1.2% gains). To maximize the algorithmic benefits, we propose minor extensions to contemporary CNN accelerators, which can be integrated into a mobile Systems-on-a-Chip (SoC) without modifying other SoC components. With additional hardware support, Mesorasi achieves up to 3.6x speedup.

cs.CV↗
arXiv2019

Tigris: Architecture and Algorithms for 3D Perception in Point Clouds

Machine perception applications are increasingly moving toward manipulating and processing 3D point cloud. This paper focuses on point cloud registration, a key primitive of 3D data processing widely used in high-level tasks such as odometry, simultaneous localization and mapping, and 3D reconstruction. As these applications are routinely deployed in energy-constrained environments, real-time and energy-efficient point cloud registration is critical. We present Tigris, an algorithm-architecture co-designed system specialized for point cloud registration. Through an extensive exploration of the registration pipeline design space, we find that, while different design points make vastly different trade-offs between accuracy and performance, KD-tree search is a common performance bottleneck, and thus is an ideal candidate for architectural specialization. While KD-tree search is inherently sequential, we propose an acceleration-amenable data structure and search algorithm that exposes different forms of parallelism of KD-tree search in the context of point cloud registration. The co-designed accelerator systematically exploits the parallelism while incorporating a set of architectural techniques that further improve the accelerator efficiency. Overall, Tigris achieves 77.2$\times$ speedup and 7.4$\times$ power reduction in KD-tree search over an RTX 2080 Ti GPU, which translates to a 41.7% registration performance improvements and 3.0$\times$ power reduction.

cs.CV↗