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Loop Space Decompositions of Connected Sums and Applications to the Vigu\'e-Poirrier Conjecture
Recent work of Beben and Theriault on decomposing based loop spaces of highly
connected Poincar\'e Duality complexes has yielded new methods for analysing
the homotopy theory of manifolds. In this paper we will expand upon these
methods, which we will then apply to give new examples supporting a long
standing question of rational homotopy theory: the Vigu\'e-Poirrier Conjecture.Comment: Minor changes from previous version. 16 pages, comments welcome.
Sections 2 and 3 subsume parts of arXiv:2210.04548 [math.AT
Measuring Value Understanding in Language Models through Discriminator-Critique Gap
Recent advancements in Large Language Models (LLMs) have heightened concerns
about their potential misalignment with human values. However, evaluating their
grasp of these values is complex due to their intricate and adaptable nature.
We argue that truly understanding values in LLMs requires considering both
"know what" and "know why". To this end, we present the Value Understanding
Measurement (VUM) framework that quantitatively assesses both "know what" and
"know why" by measuring the discriminator-critique gap related to human values.
Using the Schwartz Value Survey, we specify our evaluation values and develop a
thousand-level dialogue dataset with GPT-4. Our assessment looks at both the
value alignment of LLM's outputs compared to baseline answers and how LLM
responses align with reasons for value recognition versus GPT-4's annotations.
We evaluate five representative LLMs and provide strong evidence that the
scaling law significantly impacts "know what" but not much on "know why", which
has consistently maintained a high level. This may further suggest that LLMs
might craft plausible explanations based on the provided context without truly
understanding their inherent value, indicating potential risks
SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training
In an era where symbolic mathematical equations are indispensable for
modeling complex natural phenomena, scientific inquiry often involves
collecting observations and translating them into mathematical expressions.
Recently, deep learning has emerged as a powerful tool for extracting insights
from data. However, existing models typically specialize in either numeric or
symbolic domains, and are usually trained in a supervised manner tailored to
specific tasks. This approach neglects the substantial benefits that could
arise from a task-agnostic unified understanding between symbolic equations and
their numeric counterparts. To bridge the gap, we introduce SNIP, a
Symbolic-Numeric Integrated Pre-training, which employs joint contrastive
learning between symbolic and numeric domains, enhancing their mutual
similarities in the pre-trained embeddings. By performing latent space
analysis, we observe that SNIP provides cross-domain insights into the
representations, revealing that symbolic supervision enhances the embeddings of
numeric data and vice versa. We evaluate SNIP across diverse tasks, including
symbolic-to-numeric mathematical property prediction and numeric-to-symbolic
equation discovery, commonly known as symbolic regression. Results show that
SNIP effectively transfers to various tasks, consistently outperforming fully
supervised baselines and competing strongly with established task-specific
methods, especially in few-shot learning scenarios where available data is
limited
A Note on a Result of Makowski
In this note, we fix a gap in a proof of the first author that 28 is the only
even perfect number which is the sum of two perfect cubes. We also discuss the
situation for higher powers
Higher-order Topological Insulators and Semimetals in Three Dimensions without Crystalline Counterparts
Quasicrystals allow for symmetries that are impossible in crystalline
materials, such as eight-fold rotational symmetry, enabling the existence of
novel higher-order topological insulators in two dimensions without crystalline
counterparts. However, it remains an open question whether three-dimensional
higher-order topological insulators and Weyl-like semimetals without
crystalline counterparts can exist. Here, we demonstrate the existence of a
second-order topological insulator by constructing and exploring a
three-dimensional model Hamiltonian in a stack of Ammann-Beenker tiling
quasicrystalline lattices. The topological phase has eight chiral hinge modes
that lead to quantized longitudinal conductances of . We show that the
topological phase is characterized by the winding number of the quadrupole
moment. We further establish the existence of a second-order topological
insulator with time-reversal symmetry, characterized by a
topological invariant. Finally, we propose a model that exhibits a higher-order
Weyl-like semimetal phase, demonstrating both hinge and surface Fermi arcs. Our
findings highlight that quasicrystals in three dimensions can give rise to
higher-order topological insulators and semimetal phases that are unattainable
in crystals.Comment: 7 pages, 6 figure
3D-GPT: Procedural 3D Modeling with Large Language Models
In the pursuit of efficient automated content creation, procedural
generation, leveraging modifiable parameters and rule-based systems, emerges as
a promising approach. Nonetheless, it could be a demanding endeavor, given its
intricate nature necessitating a deep understanding of rules, algorithms, and
parameters. To reduce workload, we introduce 3D-GPT, a framework utilizing
large language models~(LLMs) for instruction-driven 3D modeling. 3D-GPT
positions LLMs as proficient problem solvers, dissecting the procedural 3D
modeling tasks into accessible segments and appointing the apt agent for each
task. 3D-GPT integrates three core agents: the task dispatch agent, the
conceptualization agent, and the modeling agent. They collaboratively achieve
two objectives. First, it enhances concise initial scene descriptions, evolving
them into detailed forms while dynamically adapting the text based on
subsequent instructions. Second, it integrates procedural generation,
extracting parameter values from enriched text to effortlessly interface with
3D software for asset creation. Our empirical investigations confirm that
3D-GPT not only interprets and executes instructions, delivering reliable
results but also collaborates effectively with human designers. Furthermore, it
seamlessly integrates with Blender, unlocking expanded manipulation
possibilities. Our work highlights the potential of LLMs in 3D modeling,
offering a basic framework for future advancements in scene generation and
animation.Comment: Project page: https://chuny1.github.io/3DGPT/3dgpt.htm
Variational Inference for SDEs Driven by Fractional Noise
We present a novel variational framework for performing inference in (neural)
stochastic differential equations (SDEs) driven by Markov-approximate
fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling
real-world continuous-time dynamic systems with inherent noise and randomness.
Combining SDEs with the powerful inference capabilities of variational methods,
enables the learning of representative function distributions through
stochastic gradient descent. However, conventional SDEs typically assume the
underlying noise to follow a Brownian motion (BM), which hinders their ability
to capture long-term dependencies. In contrast, fractional Brownian motion
(fBM) extends BM to encompass non-Markovian dynamics, but existing methods for
inferring fBM parameters are either computationally demanding or statistically
inefficient. In this paper, building upon the Markov approximation of fBM, we
derive the evidence lower bound essential for efficient variational inference
of posterior path measures, drawing from the well-established field of
stochastic analysis. Additionally, we provide a closed-form expression to
determine optimal approximation coefficients. Furthermore, we propose the use
of neural networks to learn the drift, diffusion and control terms within our
variational posterior, leading to the variational training of neural-SDEs. In
this framework, we also optimize the Hurst index, governing the nature of our
fractional noise. Beyond validation on synthetic data, we contribute a novel
architecture for variational latent video prediction,-an approach that, to the
best of our knowledge, enables the first variational neural-SDE application to
video perception.Comment: 24 pages, under revie
End-to-End Delay Minimization based on Joint Optimization of DNN Partitioning and Resource Allocation for Cooperative Edge Inference
Cooperative inference in Mobile Edge Computing (MEC), achieved by deploying
partitioned Deep Neural Network (DNN) models between resource-constrained user
equipments (UEs) and edge servers (ESs), has emerged as a promising paradigm.
Firstly, we consider scenarios of continuous Artificial Intelligence (AI) task
arrivals, like the object detection for video streams, and utilize a serial
queuing model for the accurate evaluation of End-to-End (E2E) delay in
cooperative edge inference. Secondly, to enhance the long-term performance of
inference systems, we formulate a multi-slot stochastic E2E delay optimization
problem that jointly considers model partitioning and multi-dimensional
resource allocation. Finally, to solve this problem, we introduce a
Lyapunov-guided Multi-Dimensional Optimization algorithm (LyMDO) that decouples
the original problem into per-slot deterministic problems, where Deep
Reinforcement Learning (DRL) and convex optimization are used for joint
optimization of partitioning decisions and complementary resource allocation.
Simulation results show that our approach effectively improves E2E delay while
balancing long-term resource constraints.Comment: 7 pages, 9 figures, 1 table, 1 algorithm, to be published in IEEE
98th Vehicular Technology Conference (VTC2023-Fall
Exploring Light-Cone Distribution Amplitudes from Quantum Computing
Light-cone distribution amplitudes (LCDAs) are essential nonperturbative
quantities for theoretical predictions of exclusive high-energy processes in
quantum chromodynamics (QCD). We demonstrate the prospect of calculating LCDAs
on a quantum computer by applying a recently proposed quantum algorithm, with
staggered fermions, to the simulation of the LCDA in the (1+1)-dimensional
Nambu-Jona-Lasinio (NJL) model on classical hardware. The agreement between the
result from the classical simulation of the quantum algorithm and that from
exact diagonalization justifies the proposed quantum algorithm. We find that
the resulting LCDA in the NJL model exhibits features shared with the LCDAs
obtained from QCD.Comment: 8 pages, 7 figures, published version in Sci.China Phys.Mech.Astro
Quantifying magnetic field driven lattice distortions in kagome metals at the femto-scale using scanning tunneling microscopy
A wide array of unusual phenomena has recently been uncovered in kagome
solids. The charge density wave (CDW) state in the kagome superconductor AV3Sb5
in particular intrigued the community -- the CDW phase appears to break the
time-reversal symmetry despite the absence of spin magnetism, which has been
tied to exotic orbital loop currents possibly intertwined with magnetic field
tunable crystal distortions. To test this connection, precise determination of
the lattice response to applied magnetic field is crucial, but can be
challenging at the atomic-scale. We establish a new scanning tunneling
microscopy based method to study the evolution of the AV3Sb5 atomic structure
as a function of magnetic field. The method substantially reduces the errors of
typical STM measurements, which are at the order of 1% when measuring an
in-plane lattice constant change. We find that the out-of-plane lattice
constant of AV3Sb5 remains unchanged (within 10^-6) by the application of both
in-plane and out-of-plane magnetic fields. We also reveal that the in-plane
lattice response to magnetic field is at most at the order of 0.05%. Our
experiments provide further constraints on time-reversal symmetry breaking in
kagome metals, and establish a new tool for higher-resolution extraction of the
field-lattice coupling at the nanoscale applicable to other quantum materials