arXiv subjects

Nathan Ng

Publications and source records attributed to Nathan Ng.

At least 37 records · Page 2Linked to original sources

arXiv2022

Real time evolution of Anderson impurity models via tensor network influence functionals

In this work we present and analyze two tensor network-based influence functional approaches for simulating the real-time dynamics of quantum impurity models such as the Anderson model. Via comparison with recent numerically exact simulations, we show that such methods accurately capture the long-time non-equilibrium quench dynamics. The two parameters that must be controlled in these tensor network influence functional approaches are a time discretization (Trotter) error and a bond dimension (tensor network truncation) error. We show that the actual numerical uncertainties are controlled by an intricate interplay of these two approximations which we demonstrate in different regimes. Our work opens the door to using these tensor network influence functional methods as general impurity solvers.

cond-mat.str-el↗

arXiv2022

Mean values of long Dirichlet polynomials with higher divisor coefficients

In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a consequence of this work, we prove special cases of conjectures of Conrey-Keating on mean values of long Dirichlet polynomials with higher order shifted divisor functions as coefficients.

math.NT↗

arXiv2022

If Influence Functions are the Answer, Then What is the Question?

Influence functions efficiently estimate the effect of removing a single training data point on a model's learned parameters. While influence estimates align well with leave-one-out retraining for linear models, recent works have shown this alignment is often poor in neural networks. In this work, we investigate the specific factors that cause this discrepancy by decomposing it into five separate terms. We study the contributions of each term on a variety of architectures and datasets and how they vary with factors such as network width and training time. While practical influence function estimates may be a poor match to leave-one-out retraining for nonlinear networks, we show they are often a good approximation to a different object we term the proximal Bregman response function (PBRF). Since the PBRF can still be used to answer many of the questions motivating influence functions, such as identifying influential or mislabeled examples, our results suggest that current algorithms for influence function estimation give more informative results than previous error analyses would suggest.

cs.LG↗

arXiv2022

Shifted moments of the Riemann zeta function

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$-th moments of the Riemann zeta function on the critical line.

math.NT↗

arXiv2022

The eighth moment of the Riemann zeta function

In this article, we establish an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture. This builds on the work of the first author on the sixth moment of the Riemann zeta function and work of Conrey-Gonek and Ivić. A key input is a sharp bound for a certain shifted moment of the Riemann zeta function, assuming the Riemann hypothesis.

math.NT↗

arXiv2022

Long-time memory effects in a localizable central spin problem

We study the properties of the Nakajima-Zwanzig memory kernel for a qubit immersed in a many-body localized (i.e., disordered and interacting) bath. We argue that the memory kernel decays as a power law in both the localized and ergodic regimes, and show how this can be leveraged to extract $t\to\infty$ populations for the qubit from finite time ($J t \leq 10^2$) data in the thermalizing phase. This allows us to quantify how the long-time values of the populations approach the expected thermalized state as the bath approaches the thermodynamic limit. This approach should provide a good complement to state-of-the-art numerical methods, for which the long-time dynamics with large baths are impossible to simulate in this phase. Additionally, our numerics on finite baths reveal the possibility for unbounded exponential growth in the memory kernel, a phenomenon rooted in the appearance of exceptional points in the projected Liouvillian governing the reduced dynamics. In small systems amenable to exact numerics, we find that these pathologies may have some correlation with delocalization.

cond-mat.dis-nn↗

arXiv2021

Note: Nonuniqueness of generalized quantum master equations for a single observable

When deriving exact generalized master equations for the evolution of a reduced set of degrees of freedom, one is free to choose what quantities are relevant by specifying projection operators. However, obtaining a reduced description does not always need to be achieved through projections--one can also use conservation laws for this purpose. Such an operation should be considered as distinct from any kind of projection; that is, projection onto a single observable yields a different form of master equation compared to that resulting from a projection followed by the application of a constraint. We give a simple example to show this point and give relationships that the different memory kernels must satisfy to yield the same dynamics.

quant-ph↗

arXiv2021

The sixth moment of the Riemann zeta function and ternary additive divisor sums

Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of the Riemann zeta function on the critical line. This provides a rigorous proof for a heuristic argument of Conrey and Gonek. Furthermore, this gives some evidence towards a conjecture of Conrey, Keating, Farmer, Rubinstein, and Snaith on shifted moments of the Riemann zeta function. In addition, this improves on a theorem of Ivic, who obtained an upper bound for the the sixth moment of the zeta function, based on the assumption of a conjectural formula for correlation sums of the triple divisor function.

math.NT↗

arXiv2021

Localization dynamics in a centrally coupled system

In systems where interactions couple a central degree of freedom and a bath, one would expect signatures of the bath's phase to be reflected in the dynamics of the central degree of freedom. This has been recently explored in connection with many-body localized baths coupled with a central qubit or a single cavity mode -- systems with growing experimental relevance in various platforms. Such models also have an interesting connection with Floquet many-body localization via quantizing the external drive, although this has been relatively unexplored. Here we adapt the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method, a well-known tree tensor network algorithm, to numerically simulate the dynamics of a central degree of freedom, represented by a $d$-level system (qudit), coupled to a disordered interacting 1D spin bath. ML-MCTDH allows us to reach $\approx 10^2$ lattice sites, a far larger system size than what is feasible with exact diagonalization or kernel polynomial methods. From the intermediate time dynamics, we find a well-defined thermodynamic limit for the qudit dynamics upon appropriate rescaling of the system-bath coupling. The spin system shows similar scaling collapse in the Edward-Anderson spin glass order parameter or entanglement entropy at relatively short times. At longer time scales, we see slow growth of the entanglement, which may arise from dephasing mechanisms in the localized system or long-range interactions mediated by the central degree of freedom. Similar signs of localization are shown to appear as well with unscaled system-bath coupling.

cond-mat.dis-nn↗

arXiv2021

Explicit zero density for the Riemann zeta function

Let $N(σ,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $σ$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(σ,T)$ commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result $N(σ,T)=\mathcal{O} ( T^{\frac83(1-σ)} (\log T)^5 )$. Ramaré recently proved an explicit version of this estimate. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants.

math.NT↗

arXiv2021

Sharper Bounds for the Chebyshev function $θ(x)$

In this article, we provide explicit bounds for the prime counting function $θ(x)$ in all ranges of $x$. The bounds for the error term for $θ(x)- x$ are of the shape $εx$ and $\frac{c_k x}{(\log x)^k}$, for $k=1,\ldots,5$. Tables of values for $ε$ and $c_k$ are provided.

math.NT↗

arXiv2020

SSMBA: Self-Supervised Manifold Based Data Augmentation for Improving Out-of-Domain Robustness

Models that perform well on a training domain often fail to generalize to out-of-domain (OOD) examples. Data augmentation is a common method used to prevent overfitting and improve OOD generalization. However, in natural language, it is difficult to generate new examples that stay on the underlying data manifold. We introduce SSMBA, a data augmentation method for generating synthetic training examples by using a pair of corruption and reconstruction functions to move randomly on a data manifold. We investigate the use of SSMBA in the natural language domain, leveraging the manifold assumption to reconstruct corrupted text with masked language models. In experiments on robustness benchmarks across 3 tasks and 9 datasets, SSMBA consistently outperforms existing data augmentation methods and baseline models on both in-domain and OOD data, achieving gains of 0.8% accuracy on OOD Amazon reviews, 1.8% accuracy on OOD MNLI, and 1.4 BLEU on in-domain IWSLT14 German-English.

cs.CL↗

arXiv2019

Inclusive prime number races

Let $π(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $π(x;q,a_1) > π(x;q,a_2) > \cdots > π(x;q,a_r)$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-Rényi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, and if the logarithmic density of the set of such $x$ exists and is positive, the prime number race is called inclusive. In breakthrough research, Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet $L$-functions. We show that the same conclusion can be reached assuming the generalized Riemann hypothesis and a substantially weaker linear independence hypothesis. In fact, we can assume that almost all of the zeros may be involved in $\mathbb{Q}$-linear relations; and we can also conclude more strongly that the associated limiting distribution has mass everywhere. This work makes use of a number of ideas from probability, the explicit formula from number theory, and the Kronecker-Weyl equidistribution theorem.

math.NT↗

arXiv2019

Simple and Effective Noisy Channel Modeling for Neural Machine Translation

Previous work on neural noisy channel modeling relied on latent variable models that incrementally process the source and target sentence. This makes decoding decisions based on partial source prefixes even though the full source is available. We pursue an alternative approach based on standard sequence to sequence models which utilize the entire source. These models perform remarkably well as channel models, even though they have neither been trained on, nor designed to factor over incomplete target sentences. Experiments with neural language models trained on billions of words show that noisy channel models can outperform a direct model by up to 3.2 BLEU on WMT'17 German-English translation. We evaluate on four language-pairs and our channel models consistently outperform strong alternatives such right-to-left reranking models and ensembles of direct models.

cs.CL↗

arXiv2019

Facebook FAIR's WMT19 News Translation Task Submission

This paper describes Facebook FAIR's submission to the WMT19 shared news translation task. We participate in two language pairs and four language directions, English <-> German and English <-> Russian. Following our submission from last year, our baseline systems are large BPE-based transformer models trained with the Fairseq sequence modeling toolkit which rely on sampled back-translations. This year we experiment with different bitext data filtering schemes, as well as with adding filtered back-translated data. We also ensemble and fine-tune our models on domain-specific data, then decode using noisy channel model reranking. Our submissions are ranked first in all four directions of the human evaluation campaign. On En->De, our system significantly outperforms other systems as well as human translations. This system improves upon our WMT'18 submission by 4.5 BLEU points.

cs.CL↗

arXiv2019

Many body localization in the presence of a central qudit

We consider a many-body localized system coupled globally to a central $d$-level system. Under an appropriate scaling of $d$ and $L$, we find evidence that the localized phase survives. We argue for two possible thermalizing phases, depending on whether the qudit becomes fully ergodic. This system provides one of the first examples of many-body localization in the presence of long-range (non-confining) interactions.

cond-mat.dis-nn↗

arXiv2019

Embryo staging with weakly-supervised region selection and dynamically-decoded predictions

To optimize clinical outcomes, fertility clinics must strategically select which embryos to transfer. Common selection heuristics are formulas expressed in terms of the durations required to reach various developmental milestones, quantities historically annotated manually by experienced embryologists based on time-lapse EmbryoScope videos. We propose a new method for automatic embryo staging that exploits several sources of structure in this time-lapse data. First, noting that in each image the embryo occupies a small subregion, we jointly train a region proposal network with the downstream classifier to isolate the embryo. Notably, because we lack ground-truth bounding boxes, our we weakly supervise the region proposal network optimizing its parameters via reinforcement learning to improve the downstream classifier's loss. Moreover, noting that embryos reaching the blastocyst stage progress monotonically through earlier stages, we develop a dynamic-programming-based decoder that post-processes our predictions to select the most likely monotonic sequence of developmental stages. Our methods outperform vanilla residual networks and rival the best numbers in contemporary papers, as measured by both per-frame accuracy and transition prediction error, despite operating on smaller data than many.

cs.CV↗

arXiv2019

fairseq: A Fast, Extensible Toolkit for Sequence Modeling

fairseq is an open-source sequence modeling toolkit that allows researchers and developers to train custom models for translation, summarization, language modeling, and other text generation tasks. The toolkit is based on PyTorch and supports distributed training across multiple GPUs and machines. We also support fast mixed-precision training and inference on modern GPUs. A demo video can be found at https://www.youtube.com/watch?v=OtgDdWtHvto

cs.CL↗