SPACETIME METRIC FROM QUANTUM–GRAVITY
CORRECTED FEYNMAN PROPAGATORS

P. Fernández de Córdoba1, J.M. Isidro2 and Rudranil Roy3
Instituto Universitario de Matemática Pura y Aplicada,
Universitat Politècnica de València, Valencia 46022, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract Differentiation of the scalar Feynman propagator with respect to the
spacetime coordinates yields the metric on the background spacetime that the
scalar particle propagates in. Now Feynman propagators can be modified in
order to include quantum–gravity corrections as induced by a zero–point length
$L>0$. These corrections cause the length element $\sqrt{s^{2}}$ to be
replaced with $\sqrt{s^{2}+4L^{2}}$ within the Feynman propagator. In this
paper we compute the metrics derived from both the quantum–gravity free
propagators and from their quantum–gravity corrected counterparts. We verify
that the latter propagators yield the same spacetime metrics as the former,
provided one measures distances greater than the quantum of length $L$. We
perform this analysis in the case of the background spacetime $\mathbb{R}^{D}$
in the Euclidean sector.

###### Contents

  1. 1 Introduction
  2. 2 The metric on $\mathbb{R}^{D}$ from Feynman propagators
    1. 2.1 Propagators
    2. 2.2 Summary of results
      1. 2.2.1 $L=0$
      2. 2.2.2 $L>0$
    3. 2.3 Discussion
  3. 3 Appendix: computation of the Feynman bitensor
    1. 3.1 $D=2$ with $L=0$
    2. 3.2 $D=2$ with $L>0$
    3. 3.3 $D=2k$ with $k>1$ and $L=0$
    4. 3.4 $D=2k$ with $k>1$ and $L>0$
    5. 3.5 $D=2k+1$ with $k\geq 1$ and $L=0$
    6. 3.6 $D=2k+1$ with $k\geq 1$ and $L>0$

##  1 Introduction

The search for a theory of quantum gravity is an ongoing effort that has kept
theoreticians busy for almost a century by now [11]. While a full–fledged
theory of quantum gravity still remains elusive (see, e.g., ref. [12] for a
recent review), under certain assumptions there are some quantum–gravity
corrections that can be more or less readily computed. For example, it is
generally believed that the existence of a quantum of length $L>0$ must be a
universal feature of quantum gravity in the mesoscopic regime [1, 2, 7, 8, 13,
14, 15, 16, 17, 18, 19, 22].

Endowing a $D$–dimensional manifold ${\cal M}$ with a Riemannian metric $g$,

${\rm d}s^{2}=g_{ij}(x){\rm d}x^{i}{\rm d}x^{j},$ (1)

permits one to consider the classical motion of a particle on the spacetime
${\cal M}$. For example a free, massive scalar has an action integral defined
as (the mass $m$ times) the proper length, $mS(x,y)=m\int_{y}^{x}{\rm d}s$.
The corresponding quantum particle is governed by the Feynman propagator

$G_{D}(x,y)=\sum_{\rm paths}\exp\left[-mS(x,y)\right].$ (2)

The propagator $G_{D}(x,y)$ depends on the points $x,y$ through the square of
the geodesic distance $s^{2}$ between $x,y$. Let $G_{D}=f(s^{2})$ formally
denote the functional expression for the Feynman propagator as a certain
function of $s^{2}$ (see, e.g., Eq. (13) below). Assume that one can invert it
in order to solve for $s^{2}$ as a function of the propagator, and let us
express this inverse relation as $s^{2}=f^{-1}(G_{D})$. Since by Eq. (1) we
have

$g_{ij}(x)=\frac{\partial^{2}}{\partial x^{i}\partial x^{j}}s^{2}(x),$ (3)

we can formally write

$g_{ij}(x)=\colon\lim_{y\to x}\frac{\partial^{2}}{\partial x^{i}\partial
y^{j}}f^{-1}\left(G_{D}(x,y)\right)\colon.$ (4)

Above, provision has been made for the fact that the Feynman propagator
$G_{D}(x,y)$ is singular when $x=y$. So one first splits the points $x$ and
$y$ a short distance apart, next one differentiates with respect to $x$ and
$y$, finally one lets $y\to x$; a procedure which is reminiscent of certain
regularisation prescriptions in quantum field theory. The colons :: in Eq. (4)
remind us that some prescription must be adopted in order to extract a finite
answer from the singularity. We will explain our prescription in due time.

Although in principle the procedure just laid out allows one to derive the
metric $g_{ij}$ from a knowledge of the Feynman propagator $G_{D}(x,y)$, in
practice there is little hope of ever being able to compute the inverse
function $s^{2}=f^{-1}(G_{D})$ explicitly. This notwithstanding, some
prescriptions to derive geometric objects from quantum–mechanical quantities
have been proposed in the literature [20, 21], that circumvent the
computational difficulty of Eq. (4). One of the aims of this paper is to put
forward an alternative prescription that, while computationally tractable,
yields back the correct spacetime metric.

We first define the Feynman bitensor $F_{ij}(x,y)$ [10]

$F_{ij}(x,y)=N_{D}\frac{\partial^{2}}{\partial x^{i}\partial
y^{j}}G_{D}(x,y),$ (5)

where $N_{D}$ is some dimension–dependent normalisation factor, to be
determined presently. Now $F_{ij}(x,y)$ qualifies as a bitensor because the
Feynman propagator $G_{D}(x,y)$ is a biscalar, hence its derivatives transform
as a bitensor. In the coincidence limit $y\to x$ one defines the Feynman
tensor $F_{ij}(x)$

$F_{ij}(x)=N_{D}\lim_{y\to x}\frac{\partial^{2}}{\partial x^{i}\partial
y^{j}}G_{D}(x,y).$ (6)

However, as $y\to x$, the right–hand side of (5) will contain singular terms
in a Laurent series expansion in powers of $s$, thus rendering (6) ill
defined. Assuming some regularisation has been adopted, the sought–for metric
will then be

$g_{ij}(x)=\colon F_{ij}(x)\colon=N_{D}\colon\lim_{y\to
x}\frac{\partial^{2}}{\partial x^{i}\partial y^{j}}G_{D}(x,y)\colon$ (7)

We see that, modulo irrelevant normalisation factors, $g_{ij}(x)$ in Eq. (7)
differs from $g_{ij}(x)$ in Eq. (4) by the absence of the inverse function
$f^{-1}$. To arrive at Eq. (7) we differentiate the Feynman propagator twice,
take the coincidence limit, and extract some finite term from the singular
terms in a Laurent expansion in $s$. We will see that the metric $g_{ij}$ of
the underlying spacetime manifold ${\cal M}$ will turn out to be encoded in
the coefficients multiplying those singularities. The operation of taking the
colons :: will consist in judiciously picking out the coefficient of an
appropriate singular term on the right–hand side of Eq. (6).

Eq. (7) correctly defines a symmetric, rank 2 tensor on ${\cal M}$. That it
actually defines a Riemannian metric requires proving that it is also positive
definite. This latter point will be established once one specifies how the
operation of taking the colons :: actually works, i.e., what particular
coefficient in the Laurent expansion is picked. One can then a posteriori
verify that the chosen term in the Laurent expansion does indeed define a
positive–definite Riemannian metric.

We can push the previous arguments a bit further and consider modifications to
the standard Feynman propagator $G_{D}$ on ${\cal M}$. By this we mean objects
that will be constructed similarly to $G_{D}$, but with some added
requirements. For example, one might require of $G_{D}$ that it be invariant
under the duality transformation [16, 17, 18, 19]

$S\longrightarrow\frac{L^{2}}{S}.$ (8)

Above, $L$ is a quantum of length on the spacetime ${\cal M}$. Imposing the
principle of invariance under the duality (8) is one way of implementing
ultraviolet (UV) completion [3, 4, 5] in an eventual theory of quantum
gravity. In our case this is achieved as follows. One modifies the standard
path integral (2) into the manifestly duality–invariant path integral

$G_{D}^{{\rm(QG)}}(x,y)=\sum_{\rm
paths}\exp\left\\{-m\left[S(x,y)+\frac{L^{2}}{S(x,y)}\right]\right\\}.$ (9)

The superscript QG in the modified Feynman propagator (9) stands for quantum
gravity. It has been established in refs. [16, 17, 18, 19] that, in the
particular case of a massive scalar particle on ${\cal M}$, the propagator (9)
includes lowest–order, quantum–gravitational effects due to the presence of
the zero–point length $L$.

We can now substitute the modified propagator (9) into Eq. (5) to define a
quantum–gravity corrected Feynman bitensor $F^{\rm(QG)}_{ij}(x,y)$

$F^{\rm(QG)}_{ij}(x,y)=N^{\rm(QG)}_{D}\frac{\partial^{2}}{\partial
x^{i}\partial y^{j}}G_{D}^{{\rm(QG)}}(x,y),$ (10)

as well as the corresponding Feynman tensor $F^{\rm(QG)}_{ij}(x)$:

$F^{\rm(QG)}_{ij}(x)=N^{\rm(QG)}_{D}\lim_{y\to x}\frac{\partial^{2}}{\partial
x^{i}\partial y^{j}}G_{D}^{{\rm(QG)}}(x,y).$ (11)

From here one derives the quantum–gravity corrected metric

$g^{\rm(QG)}_{ij}(x)=\colon
F^{\rm(QG)}_{ij}(x)\colon=N_{D}^{\rm(QG)}\colon\lim_{y\to
x}\frac{\partial^{2}}{\partial x^{i}\partial y^{j}}G_{D}^{\rm(QG)}(x,y)\colon$
(12)

Although there is no a priori reason for $g^{\rm(QG)}_{ij}$ to coincide with
its parent metric $g_{ij}$, they will actually be the same provided one
measures distances greater than the quantum of length $L$. The reason is the
form invariance of the Feynman propagators before and after imposing
invariance under the duality transformation (8) (see Eqs. (13), (14) below).

In this work we first consider the known Feynman propagator for a free,
massive scalar particle on the Euclidean spacetime manifold $\mathbb{R}^{D}$
and validate our prescription (7) by verifying that the original metric is
retrieved as expected. Next we move on to the duality–invariant Feynman
propagator derived in ref. [16, 17, 19], compute the corresponding metric via
our prescription (12) and verify that, for distances larger than $L$, it
coincides with that obtained from the original (i.e., non duality invariant)
Feynman propagator.

For simplicity we will limit our analysis to the flat spacetime ${\cal
M}=\mathbb{R}^{D}$ endowed with the standard Euclidean metric; in upcoming
work we will analyse curved spacetimes [6]. In our use of special functions we
follow the conventions of ref. [9].

##  2 The metric on $\mathbb{R}^{D}$ from Feynman propagators

###  2.1 Propagators

A massive, free scalar particle in Euclidean space $\mathbb{R}^{D}$ has the
Feynman propagator

$G_{D}(s)=\frac{m^{D-2}}{(2\pi)^{D/2}}\frac{K_{D/2-1}\left(ms\right)}{\left(ms\right)^{D/2-1}},\qquad
s^{2}(x,y)=\sum_{i=1}^{D}(x^{i}-y^{i})^{2},$ (13)

where $K_{n}(z)$ is a modified Bessel function and $s^{2}(x,y)$ is the square
of the geodesic distance between points $x,y\in\mathbb{R}^{D}$. By homogeneity
and isotropy, $x,y$ enter the Feynman propagator only through the combination
$s$.

Corresponding to the gravity–free propagator (13) we have a duality–invariant
propagator [16, 17, 19]

$G^{\rm(QG)}_{D}(s)=\frac{m^{D-2}}{(2\pi)^{D/2}}\frac{K_{D/2-1}\left(m\sqrt{s^{2}+4L^{2}}\right)}{\left(m\sqrt{s^{2}+4L^{2}}\right)^{D/2-1}}.$
(14)

We observe that the functional form of the Feynman propagator is the same in
Eqs. (13) and (14), the only difference being the shift

$s=\sqrt{s^{2}}\longrightarrow\sqrt{s^{2}+4L^{2}}$ (15)

due to the zero–point length $L>0$.

###  2.2 Summary of results

In this section we summarise our main results; computational details are
presented in the appendix.

####  2.2.1 $L=0$

In the absence of a quantum of length one arrives at a Feynman bitensor (see
Eqs. (29), (33) and (37))

$F_{ij}(x,y)=N_{D}\frac{\partial^{2}}{\partial x^{i}\partial
y^{j}}G_{D}(x,y)=\frac{1}{s^{D}}\left[\delta_{ij}-\frac{D}{s^{2}}\left(x^{i}-y^{i}\right)\left(x^{j}-y^{j}\right)\right],$
(16)

and the normalisation factor turns out to be

$N_{D}=\frac{2\pi^{D/2}}{\Gamma(D/2)}.$ (17)

In this case the regularisation prescription (7) reads: the metric is given by
the coefficient multiplying the least singular term in the Laurent expansion
(16). That is,

$g_{ij}(x)=\colon F_{ij}(x)\colon=\delta_{ij}.$ (18)

####  2.2.2 $L>0$

Here the Feynman bitensor (16) gets replaced with (see Eqs. (31), (35) and
(39))

$F^{\rm(QG)}_{ij}(x,y)=N^{\rm(QG)}_{D}\frac{\partial^{2}}{\partial
x^{i}\partial y^{j}}G^{\rm(QG)}_{D}(x,y)$ (19)
$=\frac{1}{\left(\sqrt{s^{2}+4L^{2}}\right)^{D}}\left[\delta_{ij}-\frac{D}{s^{2}+4L^{2}}\left(x^{i}-y^{i}\right)\left(x^{j}-y^{j}\right)\right],$

while the normalisation factor remains as above:

$N^{\rm(QG)}_{D}=\frac{2\pi^{D/2}}{\Gamma(D/2)}.$ (20)

The expansion (19) is no longer singular at $s=0$ because the zero–point
length $L>0$ prevents it. Without loss of generality we can set $x=0$ and
consider the coincidence limit to be $y\to 0$:

$\lim_{y\to 0}F^{\rm(QG)}_{ij}(y)=\lim_{y\to
0}\left[\frac{1}{\left(2L\right)^{D}}\left(\delta_{ij}-\frac{D}{4L^{2}}y^{i}y^{j}\right)\right].$
(21)

However, we cannot simply set $\lim_{y\to 0}y^{i}y^{j}=0$, because that would
amount to having $L=0$. Regularisation now stands for the following
prescription: if $||{\bf y}||<L$ we take

$:\lim_{y\to 0}F^{\rm(QG)}_{ij}(y):=\delta_{ij}-\frac{D}{4L^{2}}y^{i}y^{j},$
(22)

while, if $||{\bf y}||\geq L$, we take

$:\lim_{y\to 0}F^{\rm(QG)}_{ij}(y):=\delta_{ij}.$ (23)

The overall factor $(2L)^{-D}$ plays no role and has been cancelled.
Altogether, the metric $g_{ij}^{\rm(QG)}$ reads

$g_{ij}^{\rm(QG)}(y)=\left\\{\begin{array}[]{ll}\delta_{ij}-\frac{D}{4L^{2}}y^{i}y^{j},&||{\bf
y}||<L\\\ \delta_{ij},&||{\bf y}||\geq L.\end{array}\right.$ (24)

###  2.3 Discussion

Spacetime cannot be probed at distances shorter than $L$. Hence the term
$-(D/4)y^{i}y^{j}$ correcting $\delta_{ij}$ in (24) is to be understood as
implementing a change in the metric across the interface $||{\bf y}||=L$. The
interior of the sphere $||{\bf y}||=L$ still makes sense as a mathematical
construct, but not as a description of physical reality as seen by an observer
sitting inside. To reiterate: there is no physical metric for values of ${\bf
y}$ such that $||{\bf y}||<L$. The QG–corrected metric (22) must be regarded
as being merely the coefficient multiplying the relevant term in a Laurent
expansion. The QG–corrected Feynman propagator imposes a natural cutoff that
renders the notion of a metric inside the sphere $||{\bf y}||=L$ physically
meningless.

The transition referred to above may, but need not, be a discontinuity such as
that in Eq. (24). In fact such a discontinuity could be removed altogether and
replaced with a continuous, even a differentiable transition across the sphere
$||{\bf y}||=L$, smoothly distributed over a few fractions of $L$ in length.
The key property is the existence of a transition (be it continuous or not,
smooth or not) as one traverses the interface between the two regimes; this
transition prevents an observer from accessing any information carrying
momenta larger than $1/L$.

Our prescription correctly captures the fact that, far enough from the origin,
the quantum of length $L>0$ becomes unobservable. Then the coarse–grained
description of the metric structure on $\mathbb{R}^{D}$ remains as given by
the standard Euclidean metric $\delta_{ij}$.

The above–mentioned transition is localised not just on a sphere around the
origin ${\bf y}={\bf 0}$. Rather, it occurs around every single point ${\bf
y}\in\mathbb{R}^{D}$, once one zooms in down to lengths of order $L$ from
${\bf y}$. Of course, by homogeneity of $\mathbb{R}^{D}$ we can limit our
discussion to the point ${\bf y}={\bf 0}$.

While the line element (1) qualifies as a bona fide Riemannian metric, the
inhomogeneous term $4L^{2}$ prevents one from interpreting $s^{2}+4L^{2}$ as a
quadratic form, hence also as a metric in the strict sense of the word. On the
other hand, the QG–corrected metric (24) does retain its interpretation as a
quadratic form.

##  3 Appendix: computation of the Feynman bitensor

Here we compute the Feynman bitensor (5) and its quantum–gravity corrected
counterpart (10), in order to identify the corresponding metrics. We will work
in increasing values of the dimension $D$, separating the cases $L=0$ and
$L>0$.

For use in connection with Eqs. (13) and (14) we recall the expansions

$K_{0}(z)=-\frac{1}{2}\sum_{l=0}^{\infty}\frac{1}{(l!)^{2}}\left(\frac{z}{2}\right)^{2l}\left[2\ln\left(\frac{z}{2}\right)-2\psi(l+1)\right]$
(25)

where $\psi(z)$ is the digamma function, and

$K_{n}(z)\simeq\frac{(n-1)!}{2}\left(\frac{2}{z}\right)^{n},\quad z\to 0,\quad
n=1,2,\ldots$ (26)

When the order $n$ of the Bessel function equals an integer plus one half we
have

$K_{n+1/2}(z)=\sqrt{\frac{\pi}{2z}}\,{\rm
e}^{-z}\,\sum_{l=0}^{n}\frac{(n+l)!}{l!(n-l)!(2z)^{l}}.$ (27)

Use of the previous expansions is justified because we are interested in the
behaviour of the Feynman propagator in a neighbourhood of the origin, where it
becomes singular. We will therefore keep just the leading singularity as $z\to
0$.

###  3.1 $D=2$ with $L=0$

The Feynman propagator in a neighbourhood of the origin reads, by Eqs. (13)
and (25),

$G_{2}(s)\simeq-\frac{1}{2\pi}\ln\left(\frac{m\,{\rm
e}^{\gamma}}{2}s\right),\qquad s\to 0,$ (28)

where $\gamma=-\psi(1)$ is the Euler–Mascheroni constant. A short computation
produces the Feynman bitensor

$2\pi\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial
y^{j}}G_{2}(s)=\frac{1}{s^{2}}\left[\delta_{ij}-\frac{2}{s^{2}}(x^{i}-y^{i})(x^{j}-y^{j})\right].$
(29)

###  3.2 $D=2$ with $L>0$

Next we let $L>0$. By Eqs. (14) and (25), in a neighbourhood of the origin the
Feynman propagator reads

$G_{2}^{\rm(QG)}(s)\simeq-\frac{1}{2\pi}\ln\left(\frac{m\,{\rm
e}^{\gamma}}{2}\sqrt{s^{2}+4L^{2}}\right),\qquad s\to 0,$ (30)

and for the Feynman bitensor one finds

$2\pi\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial
y^{j}}G_{2}^{\rm(QG)}(s)=\frac{1}{\left(\sqrt{s^{2}+4L^{2}}\right)^{2}}\left[\delta_{ij}-\frac{2}{s^{2}+4L^{2}}(x^{i}-y^{i})(x^{j}-y^{j})\right].$
(31)

The above correctly reduces to the Feynman bitensor (29) when $L=0$. We
observe that, as expected, the Laurent expansion (31) is no longer in powers
of $s^{2}$ but in powers of its shifted counterpart $s^{2}+4L^{2}$ instead.
This is consistent with the fact, mentioned in connection with Eq. (14), that
the functional dependence (on the line element) of the original and of the
corrected propagator is the same in both cases.

###  3.3 $D=2k$ with $k>1$ and $L=0$

By Eq. (26), the Feynman propagator (13) becomes, in a neighbourhood of the
origin,

$G_{2k}(s)\simeq\frac{(k-2)!}{4\pi^{k}}s^{2-2k},\qquad s\to 0.$ (32)

For the Feynman bitensor (6) one finds

$\frac{2\pi^{k}}{(k-1)!}\frac{\partial}{\partial
x^{i}}\frac{\partial}{\partial
y^{j}}G_{2k}(s)=\frac{1}{s^{2k}}\left[\delta_{ij}-\frac{2k}{s^{2}}\left(x^{i}-y^{i}\right)\left(x^{j}-y^{j}\right)\right].$
(33)

The above results correctly reproduce those of $D=2$ upon setting $k=1$, even
if the Bessel functions involved in the cases $k=1$ and $k>1$ have different
behaviours around the origin. For this reason they have been treated
separately.

###  3.4 $D=2k$ with $k>1$ and $L>0$

As usual we need the behaviour of the quantum–gravity corrected Feynman
propagator (14) in a neighbourhood of the origin: by (26), this is

$G^{\rm(QG)}_{2k}(s)\simeq\frac{(k-2)!}{4\pi^{k}}(s^{2}+4L^{2})^{1-k},\qquad
s\to 0.$ (34)

Using this one computes the Feynman bitensor

$\frac{2\pi^{k}}{(k-1)!}\frac{\partial}{\partial
x^{i}}\frac{\partial}{\partial y^{j}}G_{2k}^{\rm(QG)}(s)$ (35)
$=\frac{1}{(\sqrt{s^{2}+4L^{2}})^{2k}}\left[\delta_{ij}-\frac{2k}{s^{2}+4L^{2}}\left(x^{i}-y^{i}\right)\left(x^{j}-y^{j}\right)\right],$

which reduces to that in Eq. (33) upon setting $L=0$.

###  3.5 $D=2k+1$ with $k\geq 1$ and $L=0$

In this case we have, in a neighbourhood of the origin,

$G_{2k+1}(s)\simeq\frac{1}{(4\pi)^{k}}\frac{(2k-2)!}{(k-1)!}\,s^{1-2k},\qquad
s\to 0,$ (36)

and therefore

$(4\pi)^{k}\frac{(k-1)!}{(2k-1)!}\frac{\partial}{\partial
x^{i}}\frac{\partial}{\partial
y^{j}}G_{2k+1}(s)=\frac{1}{s^{2k+1}}\left[\delta_{ij}-\frac{2k+1}{s^{2}}(x^{i}-y^{i})(x^{j}-y^{j})\right].$
(37)

###  3.6 $D=2k+1$ with $k\geq 1$ and $L>0$

In a neighbourhood of the origin the Feynman propagator reads

$G_{2k+1}^{\rm(QG)}(s)\simeq\frac{1}{(4\pi)^{k}}\frac{(2k-2)!}{(k-1)!}\,\left(s^{2}+4L^{2}\right)^{-k+1/2},\quad
s\to 0.$ (38)

Proceeding as above one arrives at the Feynman bitensor

$(4\pi)^{k}\frac{(k-1)!}{(2k-1)!}\frac{\partial}{\partial
x^{i}}\frac{\partial}{\partial y^{j}}G_{2k+1}^{\rm(QG)}(s)$ (39)
$=\frac{1}{\left(\sqrt{s^{2}+4L^{2}}\right)^{2k+1}}\left[\delta_{ij}-\frac{2k+1}{s^{2}+4L^{2}}(x^{i}-y^{i})(x^{j}-y^{j})\right],$

which reduces to that in Eq. (37) upon setting $L=0$.

Acknowledgments Work funded by FEDER/MCIN under grant PID2021-128676OB-I00
(Spain).

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# Magic for the Age of Quantized DNNs

Yoshihide Sawada Equal contribution.
Preprint. Work in progress. Tokyo Research Center, AISIN, Tokyo, Japan  Ryuji
Saiin  AISIN SOFTWARE, Aichi, Japan  Kazuma Suetake  AISIN SOFTWARE, Aichi,
Japan

###### Abstract

Recently, the number of parameters in DNNs has explosively increased, as
exemplified by LLMs (Large Language Models), making inference on small-scale
computers more difficult. Model compression technology is, therefore,
essential for integration into products. In this paper, we propose a method of
quantization-aware training. We introduce a novel normalization (Layer-Batch
Normalization) that is independent of the mini-batch size and does not require
any additional computation cost during inference. Then, we quantize the
weights by the scaled round-clip function with the weight standardization. We
also quantize activation functions using the same function and apply surrogate
gradients to train the model with both quantized weights and the quantized
activation functions. We call this method Magic for the age of Quantised DNNs
(MaQD). Experimental results show that our quantization method can be achieved
with minimal accuracy degradation.

##  1 Introduction

> “
>
> Any sufficiently advanced technology is indistinguishable from magic.
>
> _Arthur C. Clarke[6]_ ”

As computational power increases, the trend toward training large-scale deep
neural networks (DNNs) on large-scale datasets continues unabated [5, 17, 31],
and the use of cloud computing is essential for such large-scale DNNs.
However, when real-time processing is required, such as in automatic driving
peripheral recognition, online processing becomes difficult with the use of
cloud computing. This implies that technology is required to integrate large-
scale DNNs into small IoT devices, such as those based on microcontroller
units [19, 20, 21]. Namely, model compression techniques, as in [13, 16, 22,
33], for integration into hardware will become more critical.

We are considering ways to quantize DNNs while training rather than quantizing
the trained model, as we expect that the quantization-aware training has a
better trade-off between compression efficiency and inference accuracy. To
achieve efficient quantization, we introduce a novel normalization technique
called Layer-Batch Normalization (LBN), which is independent of the mini-batch
size, unlike Batch Normalization (BN) [15]. LBN does not require
computationally expensive expected value processes during inference for the
normalization, unlike layer normalization (LN) [3]. In addition, because the
proposed method is independent of the mini-batch size, we can train DNNs with
smaller mini-batch sizes while maintaining accuracy. While BN recommends the
use of large mini-batch sizes for training on large-scale datasets [29], which
has resulted in an increased demand for machine resources, the proposed method
can help alleviate this resource requirement. Figure 1 represents the overview
of the LBN. On the basis of this normalization, weight standardization (WS)
[26] and quantization by scaled round-clip functions and surrogate gradients
[4, 7, 13, 30] can be combined to obtain the quantized DNNs with minimal
accuracy degradation. We call this method Magic for the age of Quantized DNNs
(MaQD).

Our contributions of this study include the following:

  1. (1)

We propose a novel normalization, LBN, which is independent of the mini-batch
size and does not require computationally expensive expected value processes
during inference.

  2. (2)

We propose the MaQD, which combines the LBN, WS, scaled round-clip functions,
and surrogate gradients, to train the quantized DNNs.

  3. (3)

Experimental results show that our quantization technique achieves a good
trade-off between compression efficiency and inference accuracy.

Figure 1: Range per normalization layer. Green areas are used for
normalization, where $C$ denotes the channel axis, $N$ denotes the batch axes,
and $H$ and $W$ denote the height and width axies, respectively. From left to
right, BN, LN, and LBN are represented, respectively.

##  2 Preliminaries

In this section, we explain the traditional normalization [3, 15] and WS [26].

###  2.1 Normalization Layer

Let $a_{i,j,k}^{(l)}\in\mathbb{R}$ be the feature after weight transformation
in the neural network, where $l\in\mathbb{N}$ represents the layer,
$i\in\mathbb{N}\ (1\leq i\leq C^{(l)})$ represents the channel,
$(j,k)\in\mathbb{N}^{2}\ (1\leq j\leq H^{(l)},1\leq k\leq W^{(l)})$ represents
the two-dimensional coordinates (because we consider the image classification
tasks), and $C^{(l)}$, $W^{(l)}$, and $H^{(l)}$ represent the channel, width,
and height of the feature map in $l$-th layer. Then, the ordinary form of the
normalization layer in the neural network, when $\mathbb{E}[\cdot]$ represents
a certain expectation operator, is defined as follows:

$\displaystyle\begin{aligned}
\hat{a}_{i,j,k}&:=\frac{g_{i,j,k}}{\sigma_{i,j,k}}(a_{i,j,k}-\mu_{i,j,k})+b_{i,j,k},\\\
\mu_{i,j,k}&:=\mathbb{E}[a_{i,j,k}],\\\
\sigma_{i,j,k}&:=\sqrt{\mathbb{E}[(a_{i,j,k}-\mu_{i,j,k})^{2}]+\epsilon},\\\
\end{aligned}$ (1)

where $g_{i,j,k},b_{i,j,k}\in\mathbb{R}$ are the trainable parameters, and
$\epsilon>0$ is the constant to prevent divide-by-zero. Note that the
normalization layer is applied to the features after the weight transformation
and before the activation function, and the symbol $(l)$ is removed for the
simplicity. In addition, note that values can be pre-merged into other
parameters (e.g., weights) if the $\mu_{i,j,k}$ and $\sigma_{i,j,k}$ are
independent of input data during inference (i.e., in the case when the values
can be fixed during inference).

####  2.1.1 Batch Normalization (BN) and Layer Normalization (LN)

In this section, we explain the LN [3] and BN [15], which are utilized by many
DNNs, including Transformer [32], ResNet [10], and DenseNet [12]. The
following equations represent the BN and LN, respectively.

$\displaystyle\begin{aligned} \mathbb{E}_{\rm
BN}[z_{i,j,k}]&:=\frac{1}{HWN}\sum_{j=1}^{H}\sum_{k=1}^{W}\sum_{n=1}^{N}[z_{i,j,k}],\\\
\end{aligned}$ (2) $\displaystyle\begin{aligned} \mathbb{E}_{\rm
LN}[z_{i,j,k}]&:=\frac{1}{CHW}\sum_{i=1}^{C}\sum_{j=1}^{H}\sum_{k=1}^{W}[z_{i,j,k}],\\\
\end{aligned}$ (3)

where $z_{i,j,k}=z_{i,j,k}(\mbox{\boldmath$x$}_{n})$ represents a function
depending on a sample $\mbox{\boldmath$x$}_{n}\ (1\leq n\leq N)$ in the mini-
batch. For the trainable parameters $g_{i,j,k}$ and $b_{i,j,k}$, the BN
provides only channel dependence, whereas the LN provides both channel and
coordinate dependence.

###  2.2 Weight Standardization (WS)

This section describes the standardization method for weights. Let $W_{{\rm
out},{\rm in}}\in\mathbb{R}$ denote the weight parameter. Then, the WS is
denoted as follows [26]:

$\displaystyle\begin{aligned} \hat{W}_{{\rm out},{\rm in}}&:=\frac{W_{{\rm
out},{\rm in}}-\mu_{W_{\rm out}}}{\sqrt{F_{\rm In}}\sigma_{W_{\rm
out}}+\epsilon},\\\ \mu_{W_{\rm out}}&:=\frac{1}{F_{\rm In}}\sum_{{\rm
in}=1}^{F_{\rm In}}W_{{\rm out},{\rm in}},\\\ \sigma_{W_{\rm
out}}&:=\sqrt{\frac{1}{F_{\rm In}}\sum_{{\rm in}=1}^{F_{\rm In}}(W_{{\rm
out},{\rm in}}-\mu_{W_{\rm out}})^{2}},\\\ \end{aligned}$ (4)

where $F_{\rm In}$ represents the number of fan-in.

##  3 Proposed Method

###  3.1 Layer-Batch Normalization (LBN)

The BN is a standard method for obtaining stable and good convergence results
when training DNNs. However, to achieve this effect, ensuring a sufficiently
large mini-batch size is necessary, which often requires a significant amount
of memory capacity. On the other hand, the LN does not cause mini-batch
dependence but does not provide the same training efficiency as the BN.

To complement these normalization layers, we propose the LBN as follows:

$\displaystyle\begin{aligned} \mathbb{E}_{\rm
LBN}[z_{i,j,k}]&:=\frac{1}{CHWN}\sum_{i=1}^{C}\sum_{j=1}^{H}\sum_{k=1}^{W}\sum_{n=1}^{N}[z_{i,j,k}].\\\
\end{aligned}$ (5)

Note that, for the trainable parameters $g_{i,j,k}$ and $b_{i,j,k}$, only the
channel dependence is given, as in the BN.

###  3.2 Magic for the age of Quantized DNNs (MaQD)

This section explains the quantization of weights and activation functions. To
quantize them, we use the following scaled round-clip function.

$\displaystyle
Q(z,\operatorname{\Delta},\operatorname{a},\operatorname{b}):=\max\left(\operatorname{a},\min\left(\operatorname{b},\frac{\operatorname{ROUND}(\operatorname{\Delta}z)}{\operatorname{\Delta}}\right)\right),$
(6)

where $z\in\mathbb{R}$ represents the input to this function,
$\operatorname{a},\operatorname{b}\in\mathbb{R}$ represent the minimum and
maximum values respectively, $\operatorname{ROUND}(\cdot)$ represents the
round function which outputs the integer nearest to the input, and
$\operatorname{\Delta}\in\mathbb{R}$ represents a value related to the number
of states. Note that this function differs from [22] in that it includes
$\operatorname{\Delta}$, which enables control over the degree of
quantization.

Using Eq. 6, qauantizations of the weights and activation functions are
carried out as follows111We can select either $M_{\rm w}/2$ or $(M_{\rm
w}-1)/2$..

$\displaystyle Q_{\rm w}(\hat{W}_{{\rm out},{\rm in}},M_{\rm w})$
$\displaystyle:=Q\left(s\hat{W}_{{\rm out},{\rm in}},\frac{M_{\rm w}}{2}\ {\rm
or}\ \frac{M_{\rm w}-1}{2},-1,1\right),$ (7) $\displaystyle Q_{\rm
a}(\hat{a}_{i,j,k},M_{\rm a})$ $\displaystyle:=Q\left(\hat{a}_{i,j,k},M_{\rm
a}-1,0,1\right),$ (8)

where $M_{\rm w}\in 2\mathbb{N}+1$ and $M_{\rm a}\in\mathbb{N}_{\geq 2}$.
$s\in\mathbb{R}$ represents the scaling factor, and we set $s=1/3$. This value
gives a confidence interval of 99.7% if the distribution of weights
corresponds to Gaussian. $M_{\rm w}$ and $M_{\rm a}$ represent the number of
states for weights and activation functions, respectively. For example, if we
quantize weights to 2 bits, we take values of $\\{-1,0,1\\}$ and set
$M_{w}=3$. This setting corresponds to 1.58 bits in [22]. Also, unlike
quantization of weights, quantization of the activation function does not
include values under zero. Namely, if we set 2 bits,
$Q_{a}(\hat{a}_{i,j,k},M_{a})$ takes values of $\\{0,1/3,2/3,1\\}$, and
$M_{a}$ becomes $4=2^{2}$. Note that we can achieve high computational
efficiency and memory savings in such a quantized model by preparing
additional floating scaling factors for each state during inference, as in
[27].

###  3.3 Training Process

In this article, we focus on the image classification tasks using the loss
function $L$ [28, 34] described as follows:

$\displaystyle\begin{aligned}
L:=(1-\gamma)\text{CE}+\gamma\text{MSE},\end{aligned}$ (9)

where CE is the cross-entropy between the label and softmax output, and MSE is
the mean-squared error between the one-hot encoded label and logit of softmax
output. $\gamma\in\mathbb{R}$ is the hyperparameter to balance two losses and
we set $\gamma=0.05$. Note that while accuracy improvement through knowledge
distillation [11, 24, 25] could easily be introduced to MaQD, we do not
address this integration in this article as our focus is solely on the
quantization itself.

To train models using the aforementioned loss function, differential
calculations are necessary. However, MaQD is clearly non-differentiable. In
this article, we adopt the surrogate gradient technique to approximate the
gradient of the weights and activation functions as follows:

$\displaystyle\frac{\partial Q_{\rm w}(\hat{W}_{{\rm out},{\rm in}},M_{\rm
w})}{\partial\hat{W}_{{\rm out},{\rm in}}}$
$\displaystyle:=1_{\left|s\hat{W}_{{\rm out},{\rm
in}}\right|<1}(s\hat{W}_{{\rm out},{\rm in}}),$ (10)
$\displaystyle\frac{\partial Q_{\rm a}(\hat{a}_{i,j,k},M_{\rm
a})}{\partial\hat{a}_{i,j,k}}$ $\displaystyle:=\sum_{m=1}^{M_{\rm
a}-1}\frac{\partial\sigma_{\alpha}(\hat{a}_{i,j,k}-b_{m})}{\partial\hat{a}_{i,j,k}},$
(11)

where $1_{X}(\cdot)$ denotes the indicator function on the set $X$, and
$\partial\sigma_{\alpha}(\hat{z}_{i,j,k}-b_{m})/\partial\hat{z}_{i,j,k}$ is
the gradient of the scaled sigmoid function [30].

$\displaystyle\frac{\partial\sigma_{\alpha}(z)}{\partial z}$
$\displaystyle=\frac{1}{\alpha}\sigma_{\alpha}(z)(1-\sigma_{\alpha}(z)),$ (12)
$\displaystyle\sigma_{\alpha}(z)$
$\displaystyle:=\frac{1}{1+\exp{(-z/\alpha)}}.$ (13)

Here, we set $\alpha=0.25$. $b_{m}\in\mathbb{R}$ in Eq. 11 represents the
threshold changing from $m$-th state to $(m+1)$-th state (e.g., the timing at
which the output changes from 0 to 1 when we quantize to 1 bit), and the
equation can be described as follows:

$\displaystyle b_{m}$
$\displaystyle=\frac{1}{\operatorname{\Delta}}\left(\operatorname{ROUND}_{m}(\operatorname{\Delta}z)+\frac{1}{2}\right),$
(14) $\displaystyle=\frac{1}{M_{\rm a}-1}\left((m-1)+\frac{1}{2}\right),$ (15)

where $m\in\mathbb{N}_{<M_{\rm a}}$, and $\operatorname{ROUND}_{m}(\cdot)$
denotes the $m$-th state (i.e., $\operatorname{ROUND}_{m}(\cdot):=m-1$).
Figure 2 shows each surrogate gradient. Note that when the surrogate gradient
of the activation function was set to Straight-Through Estimator (STE) [4, 7]
as well as weights, the accuracy deteriorates significantly.

(a) Weight

(b) Activation function

Figure 2: Surrogate gradients for weights and activation functions. The blue
line represents forward propagation while the orange line represents
backpropagation (surrogate gradient).

##  4 Experiments

We first demonstrate the effectiveness of the LBN (+WS) and then present the
quantization results of the MaQD.

###  4.1 Verification of the Efficacy of LBN

In this section, we compare the normalization techniques, i.e., LBN+WS, LBN,
BN+WS, and BN.

####  4.1.1 Settings

The following conditions were adopted to observe the differences between
different normalizations when the mini-batch size varies:

  * •

Dataset: CIFAR-100 [18]

  * •

Network architecture: Non-quantized 9-layer CNN

  * •

Hyperparameters:

    * –

Optimizer: Momentum SGD

    * –

Scheduler: Cosine Annealing

    * –

Learning rate, the number of epochs, and weight decay: The values obtained
from the search with a mini-batch size of $128$ were used as baseline values,
and for the learning rate, the baseline values were multiplied by $N/128$
depending on the mini-batch size $N$. In particular, the total number of
updates was automatically adjusted as the mini-batch size changed, while
keeping the number of epochs constant.

####  4.1.2 Results

Figure 3 shows the experimental results. From the training results of each
normalization method when varying the mini-batch size (Fig. 3, left), the
following can be observed:

  * •

LBN+WS achieves lower training losses than any of the other methods,
regardless of the mini-batch size.

  * •

LBN+WS exhibits less dependence on mini-batch size compared to BN, and loss
difference between LBN+WS and BN diminishes as the mini-batch size increases.

Next, from the GPU memory usage during training for each normalization when
varying the mini-batch size (Fig. 3, right), the following can be observed:

  * •

GPU memory usage increases in proportion to the mini-batch size.

  * •

GPU memory usage varies among the methods, with the order being
BN $<$ LBN $<$ BN+WS $<$ LBN+WS (although it may be difficult to discern from
the figure, LBN uses slightly more GPU memory than BN).

These experimental results indicate that incorporating WS into BN does not
enhance the independence of mini-batch size and that LBN without WS does not
effectively reduce training and test losses. Therefore, it is crucial to
utilize both WS and LBN in conjunction. Also, the impact on GPU memory usage
is primarily determined by the mini-batch size rather than the choice of the
normalization. Therefore, since LBN is independent of the mini-batch size, we
require less GPU memory than existing methods while achieving low training and
test losses (although LBN utilizes slightly more GPU memory than BN).

Figure 3: Left: Losses for training with varying mini-batch sizes for each
normalization method. The solid and dashed lines represent the loss for the
training and test datasets, respectively. Right: GPU memory usage when the
mini-batch size is varied for each normalization method.

###  4.2 Verification of the Efficacy of MaQD

As described above, the LBN+WS provides better results than BN, especially
when we set a small mini-batch size. Thus, this section shows the quantization
results using LBN+WS (i.e., MaQD).

####  4.2.1 Settings

The following conditions were adopted to observe the differences between
different normalizations when varying the mini-batch size:

  * •

Dataset: CIFAR-10 [18] and CIFAR-100

  * •

Network architecture: VGG and PreActResNet (see Appendix for details)

  * •

Hyperparameters:

    * –

$M_{\rm w}/2\ {\rm or}\ (M_{\rm w}-1)/2$ in Eq. 7: $M_{\rm w}/2$

    * –

Optimizer: Momentum SGD

    * –

Scheduler: Cosine Annealing

    * –

Learning rate: $10^{-2}$

    * –

The number of epochs: 300

    * –

Mini-batch size: 100

####  4.2.2 Results

Tables 1, 2, 3, and 4 and Figs. 4 and 5 show experimental results with respect
to changing $M_{\rm w}$ and $M_{\rm a}$. Here, $R_{\rm w}$ and $R_{\rm a}$
represent the ratio of non-zero parameters of weights and outputs of the
activation functions, respectively. Note that for non-zero activation
functions, we compute the value by averaging the non-zero ratio for each test
sample. Also note that when $M_{\rm a}=2$, the model becomes S3NNs (Single
Step Spiking Neural Networks) [30]. In this case, the normalization layers can
be completely removed during inference by merging them with the weights. In
other words, inference can be done using only additive operations (details are
discussed in [30]).

From these results, the following can be observed:

  * •

Setting $M_{\rm w}$ and $M_{\rm a}$ to small values reduces accuracy.

  * •

However, sometimes the value relationship is reversed (e.g., $M_{\rm w}=255$
and $M_{\rm a}=8$ v.s. $M_{\rm w}=15$ and $M_{\rm a}=8$ in Table 1).
Especially, $M_{\rm w}=255$ and $M_{\rm a}=\\{4,8\\}$ in Table 4 outperforms
the non-quantized DNN.

  * •

If $M_{\rm w}$ is fixed and $M_{\rm a}$ increases, the value of $R_{\rm w}$
remains almost the same, while the value of $R_{a}$ increases.

  * •

Remarkably, if $M_{\rm a}$ is fixed and $M_{\rm w}$ increases, the value of
$R_{\rm w}$ increases, while the value of $R_{\rm a}$ decreases (especially in
layers close to the input).

Table 1: Performance with respect to varying $M_{\rm w}$ and $M_{\rm a}$. The dataset is CIFAR-10, and the network architecture is VGG. $M_{\rm w}$ | $M_{\rm a}$ | Accuracy [%] |  $R_{\rm w}$ [%] |  $R_{\rm a}$ [%]
---|---|---|---|---
Non-quantized | 95.98 | 100 | 39.38
3 | 2 | 92.52 | 12.91 | 14.53
3 | 4 | 94.98 | 12.94 | 26.95
3 | 8 | 95.32 | 12.95 | 32.97
15 | 2 | 94.26 | 82.04 | 14.32
15 | 4 | 95.68 | 82.42 | 25.95
15 | 8 | 95.93 | 82.36 | 33.41
255 | 2 | 94.18 | 99.00 | 13.88
255 | 4 | 95.65 | 99.02 | 24.58
255 | 8 | 95.89 | 99.02 | 32.31
Table 2: Performance with respect to varying $M_{\rm w}$ and $M_{\rm a}$. The dataset is CIFAR-10, and the network architecture is PreActResNet. $M_{\rm w}$ | $M_{\rm a}$ | Accuracy [%] |  $R_{\rm w}$ [%] |  $R_{\rm a}$ [%]
---|---|---|---|---
Non-quantized | 95.64 | 100 | 38.02
3 | 2 | 93.28 | 12.39 | 13.75
3 | 4 | 94.47 | 12.55 | 25.73
3 | 8 | 94.98 | 12.69 | 31.91
15 | 2 | 94.08 | 82.22 | 13.15
15 | 4 | 95.15 | 82.34 | 26.33
15 | 8 | 95.23 | 82.20 | 33.99
255 | 2 | 93.94 | 99.01 | 11.94
255 | 4 | 95.25 | 99.02 | 23.74
255 | 8 | 95.30 | 99.01 | 31.25
Table 3: Performance with respect to varying $M_{\rm w}$ and $M_{\rm a}$. The dataset is CIFAR-100, and the network architecture is VGG. $M_{\rm w}$ | $M_{\rm a}$ | Accuracy [%] |  $R_{\rm w}$ [%] |  $R_{\rm a}$ [%]
---|---|---|---|---
Non-quantized | 76.52 | 100 | 40.89
3 | 2 | 69.19 | 12.51 | 17.35
3 | 4 | 74.02 | 12.64 | 32.45
3 | 8 | 74.85 | 12.71 | 38.54
15 | 2 | 72.27 | 82.78 | 17.10
15 | 4 | 75.44 | 82.58 | 29.23
15 | 8 | 75.66 | 82.49 | 36.33
255 | 2 | 72.73 | 99.04 | 16.75
255 | 4 | 76.08 | 99.03 | 27.51
255 | 8 | 75.86 | 99.03 | 34.47
Table 4: Performance with respect to varying $M_{\rm w}$ and $M_{\rm a}$. The dataset is CIFAR-100, and the network architecture is PreActResNet. $M_{\rm w}$ | $M_{\rm a}$ | Accuracy [%] |  $R_{\rm w}$ [%] |  $R_{\rm a}$ [%]
---|---|---|---|---
Non-quantized | 75.47 | 100 | 37.89
3 | 2 | 70.23 | 12.32 | 15.71
3 | 4 | 73.68 | 12.63 | 29.20
3 | 8 | 74.31 | 12.71 | 34.83
15 | 2 | 71.29 | 82.58 | 15.67
15 | 4 | 75.28 | 82.39 | 28.82
15 | 8 | 75.26 | 82.33 | 34.55
255 | 2 | 72.54 | 99.03 | 13.68
255 | 4 | 75.48 | 99.02 | 24.48
255 | 8 | 75.91 | 99.02 | 31.78

Figure 4: Examples of non-zero ratios for each layer when varying $M_{\rm w}$
and fixing $M_{\rm a}$. The dataset is CIFAR-10, and the network architectures
are VGG (top) and PreActResNet (bottom). Note that since the output layer is
not quantized, the graph of $R_{\rm a}$ has one less layer index.

Figure 5: Examples of non-zero ratios for each layer when varying $M_{\rm a}$
and fixing $M_{\rm w}$. The dataset is CIFAR-10, and the network architectures
are VGG (top) and PreActResNet (bottom). Note that since the output layer is
not quantized, the graph of $R_{\rm a}$ has one less layer index.

These results demonstrate the effectiveness of MaQD, particularly with $M_{\rm
w}=15$ and $M_{\rm a}=8$, the model can be trained with an accuracy
degradation of less than 1%. This means that, instead of setting the
activation functions to 8 bits as in BitNet b1.58 [22], 3 bits may be
sufficient. Also, as mentioned above, when $M_{\rm a}=2$, only additive
operations can be used for inference. In this case, if we can utilize
neuromorphic chips like TrueNorth [2] or Loihi [8], we can take advantage of
asynchronous processing and significantly reduce power consumption [30]. That
is, the appropriate degrees for quantization should be determined while
considering the hardware for inference.

##  5 Conclusion

In this article, we proposed MaQD (Magic for the Age of Quantized DNNs) based
on LBN (Layer-Batch Normalization), the scaled round-clip function, and the
surrogate gradient. This enables model compression for product integration
while reducing machine resource requirements by employing training with small
mini-batch sizes. Applying this method to the image classification tasks, we
confirmed its effectiveness.

In the future, the system will be demonstrated in challenging settings, such
as the ImageNet dataset [9] and LLMs [5], which impose more demanding machine
resource requirements, to showcase its versatility. It could also be a helpful
combination with technologies like GaLore [35] and Evolutionary Model Merging
[1], which have the potential to accelerate the democratization of AI
development. Additionally, applications to SNNs (Spiking Neural Networks) [14,
23] that require time series processing will be explored.

We hope that ubiquitous magic eventually becomes a gift to civilization.

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##  Appendix A Network Details

In this paper, the VGG and PreActResNet used in the quantization experiments
were set up as follows.

VGG:
(64C3)$\times$2-(128C3)$\times$2-AP-(256C3)$\times$4-AP-(512C3)$\times$4-AP-(512C3)$\times$4-10C1-GAP,
PreActResNet: 64B1-128B1-256B2-256B2-512B2-512B2-512B2-512B2-10C1-GAP,

where $x$C$y$ represents the convolutional layer with $x$-output channels and
$y$-stride, AP represents the average pooling with the kernel size 2, and GAP
represents the global average pooling. Also, $x$B$y$ represents the residual
block,

$\displaystyle\begin{aligned} xBy:=xSy+xFy,\end{aligned}$ (16)

where

$\displaystyle\begin{aligned} xSy&:=\text{LBN-ACT-}x\text{C}y\text{-LBN-
ACT-}x\text{C}y\text{-LBN},\\\ xFy&:=\text{LBN-
ACT-}x\text{C}y\text{-LBN},\end{aligned}$ (17)

and ACT represents the (quantized) activation function. Note that the above
description of VGG does not show the normalization layer for the simplicity,
although the LBN is applied to the features after the weight transformation.
The activation function (not shown above) is then utilized. Also, the
PreActResNet performs AP before processing in blocks where the stride is set
to 2 (although this is not explicitly specified too).


# Stellar flybies within 1 ly from the Sun and stars passing through the Hills
cloud

Igor Yu. Potemine
Institut de Mathématiques, Université Paul Sabatier, Toulouse, France

###### Abstract

We reexamine the SIMBAD database with incorporated Gaia DR3 parallaxes and
proper motions. Appropriate query searches allow us to find several nearby
stars with measured radial velocities having flybies within 1 ly from the Sun
(in linear approximation). The closest past flyby $\approx 215.7$ kyr ago at
$\approx 0.136$ pc is attributed to the star UCAC4 323-037188, currently
located at 21.74 pc from the Sun. If the radial velocity of a star is unknown,
we attribute to it an “advantageous” value of $\pm 50$ km/s. It allows us to
create an additional list of “Nemesis candidates” for the perforation of the
inner Oort cloud. The closest potential flyby at $\approx 0.015$ pc ($\approx$
0.050 ly $\approx$ 3147 AU) from the Sun is attributed to GALEX
J013712.5-012958 (Gaia DR3 2508809660245711360). It is currently located at
the distance of about 111.1 pc and belongs to the Pisces-Eridanus stream.
Also, close flybies of some massive bright stars (Algol, A-giant HD 107914,
A-subgiant HD 2733 and B-subgiant HD 165704) are analyzed. Finally, we notice
that B1-star BD+60 596, whose proper motions and radial velocity are both
small, can be considered as the Sun’s pseudo-stream companion.

Keywords: Hills cloud, nearby star, Oort cloud, proper motion, radial
velocity, star flyby

###### Contents

  1. 1 Introduction
  2. 2 Minimal distances and times of closest approaches
  3. 3 Parametric frames for SIMBAD query searches
  4. 4 Closest stellar flybies within 1 ly
  5. 5 Extended list of Nemesis candidates
  6. 6 Close flybies of massive bright stars
  7. 7 Hypothetical pseudo-stream companion BD+60 596
  8. 8 Conclusion

##  1 Introduction

In 1916 an American astronomer E. Barnard discovered that the star BD$+04$
3561a has an extraordinary high proper motion of 10.3 arcseconds per year. It
is literally flying in the sky.

There is also a different phenomenon of stars with small proper motions but
high radial velocities. Their trajectories are almost parallel to the Sun’s
trajectory. In addition, these stars furtively approach the Sun and then
quickly run away.

Many authors (cf. [1, 2, 3, 4]) have searched past and future stellar
perturbers of the Oort cloud. Indeed, it turns out that a massive star passing
within 1 ly could have a significant influence on long-period comets. More
closely, at 10000 AU, such a star would have a serious direct influence on the
inner Oort cloud and trans-Neptunian objects.

The most known example is the K7-star Gliese 710 with closest approach at
$10635\pm 500$ AU in 1.29 Myr [5]. In recent papers (cf. [6, 7, 8]), there
appeared also the Scholz’s star and G3-star HD 7977.

In the paper [9], we examined an additional Nemesis candidate HD 107914 / HIP
60503 with very small proper motions. New results for this remarkable A7/8
giant are presented in section 6.

In the present article, we establish an updated list of Nemesis candidates and
Hills cloud perturbers.

##  2 Minimal distances and times of closest approaches

We use Julian years and light-years so that 1 parsec $=p\approx 3.261563777$
ly and 1 km/s corresponds to the velocity of $c^{-1}$ ly/y where $c\approx
299792.458$ is the speed of light in km/s.

Let $X$ be a star with parallax $\pi_{X}$ (in mas) at the current distance of
$d_{X}=p\cdot 10^{3}/\pi_{X}$ ly from the Sun denoted by the symbol $\odot$.
The minimal distance $d_{\odot X}^{\mathrm{min}}$ from $X$ to $\odot$ is equal
to the radius $R$ of the sphere centered at the Sun and tangent to the
trajectory of $X$. It is easy to show that

$\left|\frac{v_{r}}{v_{t}}\right|=\sqrt{\left(\frac{d_{X}}{R}\right)^{2}-1}$
(2.1)

where $v_{r}$ and $v_{t}$ are radial and transverse (tangential) velocities of
a star $X$ respectively. The transverse velocity can be easily calculated from
available catalogue values:

$v_{t}=\frac{\pi c}{648\cdot 10^{6}}\mu_{T}d_{X}=\frac{\pi cp}{648\cdot
10^{3}}\frac{\mu_{T}}{\pi_{X}}\approx 4.74047\cdot\frac{\mu_{T}}{\pi_{X}}\
\mathrm{km/s}$ (2.2)

where $\mu_{T}$ is the total proper motion of $X$ in mas and
$\frac{\pi}{648\cdot 10^{6}}$ the number of radians in 1 mas. From these two
formulas we obtain:

$d_{\odot X}^{\mathrm{min}}=R=\frac{d_{X}}{\sqrt{K^{2}+1}}\ \mathrm{where}\
K=\frac{v_{r}}{v_{t}}\approx\frac{v_{r}\pi_{X}}{4.74047\cdot\mu_{T}}.$ (2.3)

The time of closest approach $t_{\odot X}^{\mathrm{min}}$ in years can be
calculated, for example, by the formula:

$t_{\odot
X}^{\mathrm{min}}=\pm\frac{\sqrt{d_{X}^{2}-R^{2}}}{\sqrt{v_{r}^{2}+v_{t}^{2}}}\cdot
c.$ (2.4)

The sign is “$+$” when $v_{r}$ is negative and vice versa. It gives also a
simpler formula

$t_{\odot X}^{\mathrm{min}}=-\frac{d_{X}v_{r}}{v_{r}^{2}+v_{t}^{2}}\cdot c.$
(2.5)

##  3 Parametric frames for SIMBAD query searches

We will indicate here a systematic way to find candidates for flybies within 1
ly in the SIMBAD database.

Suppose that $R\leqslant 1$ ly. From the formulas above we get

$|K|\geqslant\sqrt{d_{X}^{2}-1}\ \textrm{and}\
v_{t}\leqslant\frac{|v_{r}|}{\sqrt{d_{X}^{2}-1}}.$ (3.1)

Thus, we obtain,

$\mu_{\,T}\lessapprox\frac{|v_{r}|\cdot\pi_{X}}{4.74047\cdot\sqrt{d_{X}^{2}-1}}.$
(3.2)

When $\pi_{X}\ll 1000$ is small enough then

$\mu_{\,T}\lessapprox\frac{|v_{r}|\cdot\pi_{X}^{2}}{4740.47\cdot p}.$ (3.3)

For example, if $\pi_{X}\approx 10$ mas and $|v_{r}|\approx 50$ km/s then
$\mu_{\,T}\lessapprox 0.324$ mas/yr.

Using formulas above, one can create an appropriate grid of key values of
$\pi_{X}$ and $v_{r}$ and make SIMBAD query searches.

##  4 Closest stellar flybies within 1 ly

All following “Nemesis candidates” can be found by a straightforward SIMBAD
search. Our results are consistent with calculations in the recent paper [8].

Table 1: List of Nemesis candidates. ID | $\pi_{X}$ | prop. mot. | $v_{t}$ | $v_{r}$
---|---|---|---|---
| (mas) | (mas/yr) | (km/s) | (km/s)
Scholz’s star | 148.8 |  $-$52.7 $-$95.8 | 3.483 |  $+$83.10
GJ 710 | 52.3963 |  $-$0.414 $-$0.108 | 0.039 |  $-$14.42
UCAC4 556-008431 | 47.8649 |  $+$0.041 $-$4.601 | 0.456 |  $+$31.08
UCAC4 323-037188 | 45.9926 |  $-$3.599 $+$4.802 | 0.619 |  $+$98.58
UCAC4 364-018255 | 25.6277 |  $-$0.328 $-$1.298 | 0.248 |  $+$45.18
HD 7977 | 13.2118 |  $+$0.144 $+$0.010 | 0.052 |  $+$26.76
UCAC4 237-008148 | 10.2340 |  $+$0.094 $+$0.461 | 0.218 |  $+$82.48

Close encounters are summarized in the following table. The closest past flyby
is attributed to the star UCAC4 323-037188, currently located at 21.74 pc from
the Sun.

Table 2: Closest stellar flybies. ID | $d_{\odot X}^{\mathrm{min}}$ | $d_{\odot X}^{\mathrm{min}}$ | $d_{\odot X}^{\mathrm{min}}$ | $t_{\odot X}^{\mathrm{min}}$
---|---|---|---|---
| (pc) | (ly) | (AU) | (kyr)
GJ 710 | 0.051 | 0.167 | 10568 |  $+$1294.1
UCAC4 323-037188 | 0.136 | 0.445 | 28138 |  $-$0215.7
HD 7977 | 0.146 | 0.478 | 30216 |  $-$2765.6
UCAC4 364-018255 | 0.214 | 0.698 | 44115 |  $-$0844.5
UCAC4 237-008148 | 0.258 | 0.842 | 53254 |  $-$1158.4
Scholz’s star | 0.281 | 0.918 | 58054 |  $-$0078.9
UCAC4 556-008431 | 0.306 | 0.999 | 63176 |  $-$0657.1

We don’t consider white dwarf candidates, like UPM J0812-3529 (cf. [10]),
having suspiciously high radial velocities, superior to $\pm 200$ km/s.

##  5 Extended list of Nemesis candidates

We use parametric frames of section 3 in order to create an extended list of
“Nemesis candidates” with unknown radial velocities, currently located within
200 pc. Stars with big measurement errors are excluded. By default, we
indicate Gaia DR3 identifiers.

Table 3: Extended list of candidates (continuing Table 1). ID | $\pi_{X}$ | proper motions | $v_{t}$
---|---|---|---
| (mas) | (mas/yr) | (km/s)
WISE J053516.80-750024.9 | 250.000 |  $-$127 $+$13 | 2.421
WISE J153541.67+222525.5 | 90.7878 |  $-$2.375 $-$19.281 | 1.014
UCAC4 767-032810 | 79.7311 |  $+$11.736 $+$12.796 | 1.032
UCAC4 638-030976 | 68.5664 |  $+$1.334 $+$0.085 | 0.092
UCAC4 563-099325 | 67.4697 |  $-$8.371 $-$4.825 | 0.679
UCAC4 486-052109 | 59.4668 |  $+$8.667 $-$7.582 | 0.918
UCAC4 563-093667 | 57.2635 |  $-$7.200 $-$1.058 | 0.602
GALEX J002332.9+432030 | 53.4296 |  $-$2.841 $-$1.834 | 0.300
4198241063381323264 | 49.8384 |  $-$1.295 $-$5.625 | 0.522
5614623799042319360 | 41.2974 |  $-$1.283 $-$4.432 | 0.530
2031432910114027648 | 41.0214 |  $+$1.439 $-$3.870 | 0.477
1835040758820804608 | 39.5958 |  $+$1.195 $-$1.283 | 0.210
UCAC4 362-032250 | 38.5171 |  $-$0.606 $+$2.903 | 0.365
4143776239543076864 | 36.6095 |  $-$0.048 $-$3.264 | 0.423
5615515773536993920 | 35.5428 |  $-$0.693 $+$3.922 | 0.531
1940843120604424576 | 35.1943 |  $-$0.196 $+$0.425 | 0.063
4158060785356113792 | 33.2454 |  $-$2.177 $-$1.184 | 0.353
5614676330771453440 | 33.2324 |  $+$0.437 $-$1.870 | 0.274
2027191994999462016 | 31.8573 |  $+$0.590 $-$2.996 | 0.454
4256872795844365568 | 31.6929 |  $-$0.575 $-$3.139 | 0.477
5698279625842983680 | 30.5893 |  $-$1.236 $-$0.576 | 0.211
2026681443634299776 | 30.2498 |  $-$0.607 $-$0.394 | 0.113
5614343011257214720 | 28.6367 |  $+$0.514 $+$1.940 | 0.332
3360979795101021824 | 27.9592 |  $+$0.947 $-$0.516 | 0.183
5615665105256804736 | 25.5830 |  $+$0.839 $+$0.434 | 0.175
5677957524207024384 | 22.0224 |  $+$0.505 $+$0.344 | 0.132
OGLE BLG-ECL-73018 | 14.6758 |  $-$0.056 $-$0.273 | 0.090
TYC 1315-1096-1 | 14.6163 |  $+$0.143 $+$0.065 | 0.051
2059704828690939136 | 12.9585 |  $-$0.055 $-$0.096 | 0.040
4113264551321565952 | 10.9226 |  $-$0.192 $-$0.296 | 0.153
4359310510300687104 | 9.9611 |  $-$0.194 $+$0.156 | 0.118
4055988791630637312 | 9.5297 |  $+$0.157 $+$0.046 | 0.081
GALEX J013712.5-012958 | 9.0019 |  $-$0.007 $+$0.011 | 0.007
4658968939769587712 | 7.5547 |  $-$0.180 $-$0.033 | 0.115
OGLE BLG-ECL-126001 | 7.3654 |  $-$0.145 $+$0.005 | 0.093
GALEX J023702.5-353520 | 6.6875 |  $+$0.086 $+$0.109 | 0.098
OGLE BLG-ECL-37494 | 5.0171 |  $-$0.010 $-$0.030 | 0.032

Potential close encounters are summarized in the following table supposing
that $|v_{r}|=50$ km/s.

Table 4: Closest stellar flybies (continuing Table 2). ID | $d_{\odot X}^{\mathrm{min}}$ | $d_{\odot X}^{\mathrm{min}}$ | $d_{\odot X}^{\mathrm{min}}$ | $t_{\odot X}^{\mathrm{min}}$
---|---|---|---|---
| (pc) | (ly) | (AU) | (kyr)
GALEX J013712.5-012958 | 0.015 | 0.050 | 3147 |  $\pm$2172.4
UCAC4 638-030976 | 0.027 | 0.088 | 5560 |  $\pm$0285.2
1940843120604424576 | 0.036 | 0.117 | 7389 |  $\pm$0555.7
2059704828690939136 | 0.062 | 0.204 | 12885 |  $\pm$1509.1
TYC 1315-1096-1 | 0.070 | 0.227 | 14379 |  $\pm$1337.9
2026681443634299776 | 0.075 | 0.245 | 15466 |  $\pm$0646.5
1835040758820804608 | 0.106 | 0.346 | 21869 |  $\pm$0493.9
GALEX J002332.9+432030 | 0.112 | 0.366 | 23164 |  $\pm$0366.0
5677957524207024384 | 0.119 | 0.390 | 24638 |  $\pm$0888.0
OGLE BLG-ECL-73018 | 0.123 | 0.400 | 25304 |  $\pm$1332.5
OGLE BLG-ECL-37494 | 0.126 | 0.411 | 26002 |  $\pm$3897.8
3360979795101021824 | 0.131 | 0.427 | 26979 |  $\pm$0699.4
5615665105256804736 | 0.137 | 0.446 | 28224 |  $\pm$0764.4
5698279625842983680 | 0.138 | 0.451 | 28499 |  $\pm$0639.3
5614676330771453440 | 0.165 | 0.538 | 34004 |  $\pm$0588.4
4055988791630637312 | 0.171 | 0.557 | 35229 |  $\pm$2052.1
UCAC4 362-032250 | 0.190 | 0.618 | 39090 |  $\pm$0507.7
WISE J053516.80-750024.9 | 0.193 | 0.631 | 39898 |  $\pm$0078.0
UCAC4 563-099325 | 0.201 | 0.656 | 41504 |  $\pm$0289.8
4198241063381323264 | 0.210 | 0.683 | 43224 |  $\pm$0392.3
UCAC4 563-093667 | 0.210 | 0.686 | 43397 |  $\pm$0341.5
4158060785356113792 | 0.213 | 0.693 | 43846 |  $\pm$0588.2
WISE J153541.67+222525.5 | 0.223 | 0.729 | 46082 |  $\pm$0215.3
4143776239543076864 | 0.231 | 0.753 | 47629 |  $\pm$0534.1
5614343011257214720 | 0.232 | 0.757 | 47858 |  $\pm$0682.9
2031432910114027648 | 0.233 | 0.759 | 47981 |  $\pm$0476.7
4359310510300687104 | 0.238 | 0.776 | 49063 |  $\pm$1963.2
OGLE BLG-ECL-126001 | 0.254 | 0.827 | 52301 |  $\pm$2655.1
5614623799042319360 | 0.256 | 0.837 | 52903 |  $\pm$0473.5
UCAC4 767-032810 | 0.259 | 0.844 | 53401 |  $\pm$0245.2
4113264551321565952 | 0.280 | 0.914 | 57833 |  $\pm$1790.4
2027191994999462016 | 0.285 | 0.930 | 58836 |  $\pm$0613.8
GALEX J023702.5-353520 | 0.294 | 0.960 | 60711 |  $\pm$2924.2
5615515773536993920 | 0.299 | 0.975 | 61650 |  $\pm$0550.1
4256872795844365568 | 0.301 | 0.982 | 62128 |  $\pm$0617.0
4658968939769587712 | 0.304 | 0.992 | 62704 |  $\pm$2588.6
UCAC4 486-052109 | 0.309 | 1.007 | 63670 |  $\pm$0328.7

When radial velocities will be measured, distances and times should be
mutiplied by $50/|v_{r}|$ approximately.

##  6 Close flybies of massive bright stars

In our previous paper [9], we examined an additional Nemesis candidate HD
107914 / HIP 60503 with small proper motions. It is a remarkable A7/8 giant,
currently at $\approx$ 79.84 pc from the Sun. Here we remake calculations
using new Gaia DR3 data.

Table 5: Astrometric data for massive bright stars. ID | $\pi_{X}$ | proper motions | $v_{t}$ | $v_{r}$
---|---|---|---|---
| (mas) | (mas/yr) | (km/s) | (km/s)
Algol | 36.27 |  $+$2.99 $-$1.66 | 0.447 |  $+$4.0
HD 107914 | 12.5254 |  $+$0.085 $+$0.966 | 0.367 |  $-$8.39
HD 165704 | 6.5505 |  $-$0.112 $+$0.011 | 0.081 |  $-$17.77
HD 2733 | 4.2927 |  $+$0.034 $+$0.009 | 0.039 |  $+$19.35

An updated search indicates also a close encounter with B9-subgiant HD 165704,
currently located at $\approx$ 152.7 pc from the Sun, and A7-subgiant HD 2733,
located at $\approx$ 233.0 pc from the Sun.

Algol ($\beta$ Persei) appears as well in the frames of section 3 with
$|v_{r}|=50$, though it’s true radial velocity is only $\approx+4$ km/s.

It gives us the following flyby data:

Table 6: Closest stellar flybies (continuing Tables 2 and 4). ID | $d_{\odot X}^{\mathrm{min}}$ | $d_{\odot X}^{\mathrm{min}}$ | $t_{\odot X}^{\mathrm{min}}$
---|---|---|---
| (pc) | (ly) | (kyr)
HD 2733 | 0.468 | 1.525 |  $-$11771.5
HD 165704 | 0.700 | 2.282 |  $+$8399.9
Algol | 3.062 | 9.986 |  $-$6656.6
HD 107914 | 3.489 | 11.380 |  $+$9286.7

##  7 Hypothetical pseudo-stream companion BD+60 596

This chemically peculiar B1-star has simultaneously small radial velocity and
proper motions. We would like to mention it despite a very high RUWE value in
Gaia DR3 catalog.

Table 7: Astrometric data for BD+60 596. ID | $\pi_{X}$ | proper motions | $v_{t}$ | $v_{r}$
---|---|---|---|---
| (mas) | (mas/yr) | (km/s) | (km/s)
BD+60 596 | 24.347 |  $-$0.743 $+$0.696 | 0.198 |  $+$1.88262

It is one of the best candidates to be a pseudo-stream companion of the Sun.

Table 8: Closest stellar flybies (continuing Tables 2, 4 and 6). ID | $d_{\odot X}^{\mathrm{min}}$ | $d_{\odot X}^{\mathrm{min}}$ | $t_{\odot X}^{\mathrm{min}}$
---|---|---|---
| (pc) | (ly) | (kyr)
BD+60 596 | 4.301 | 14.027 |  $-$21098.4

So, about 21 Myr ago, BD+60 596 was close to the Sun.

##  8 Conclusion

In this paper, we have established an extended list of stellar Nemesis
candidates. However, radial velocities of many of them are still unknown.
Also, more accurate measurements of proper motions are necessary too.

We have used just a simple linear approximation without error estimations.

This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France.

## References

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  * [3] P. Dybczyński.  Simulating observable comets III. Real stellar perturbers of the Oort cloud and their output.  A&A, 449:1233–1242, 2006.
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Igor Potemine

Institut de Mathématiques

Université Paul Sabatier

118, route de Narbonne

31062 Toulouse (France)

e-mail<EMAIL_ADDRESS>