# Invariants in Co-polar Interferometry: an Abelian Gauge Theory

Nithyanandan Thyagarajan<EMAIL_ADDRESS>https://tnithyanandan.wordpress.com CSIRO, Space & Astronomy, P. O. Box 1130,
Bentley, WA 6102, Australia  National Radio Astronomy Observatory, 1003
Lopezville Rd, Socorro, NM 87801, USA.  Rajaram Nityananda
<EMAIL_ADDRESS>Azim Premji University, Bikkanahalli Main Road,
Sarjapura, Bengaluru 562125, Karnataka, India  Joseph Samuel<EMAIL_ADDRESS>International Centre for Theoretical Sciences, Bengaluru 560089, Karnataka,
India  Raman Research Institute, Bengaluru 560080, Karnataka, India

###### Abstract

An $N$-element interferometer measures correlations among pairs of array
elements. Closure invariants associated with closed loops among array elements
are immune to multiplicative, element-based (“local”) corruptions that occur
in these measurements. Till recently, it has been unclear how a complete set
of independent invariants can be analytically determined. We view the local,
element-based corruptions in co-polar correlations as gauge tranformations
belonging to the gauge group $\textrm{GL}(1,\mathbb{C})$. Closure quantities
are then naturally gauge invariant. We use this to provide a simple and
effective formalism, and identify the complete set of independent closure
invariants from co-polar interferometric correlations using only quantities
defined on $(N-1)(N-2)/2$ elementary and independent triangular loops. The
$(N-1)(N-2)/2$ closure phases and $N(N-3)/2$ closure amplitudes (totaling
$N^{2}-3N+1$ real invariants), familiar in astronomical interferometry,
naturally emerge from this formalism, which unifies what has required separate
treatments until now. We do not require auto-correlations, but can easily
include them if reliably measured. This unified view clarifies issues relating
to noise and inference of object model parameters. It also allows us to extend
the rule of parallel transport associated with Pancharatnam phase in optics to
apply to amplitudes as well. The framework presented here extends to
$\textrm{GL}(2,\mathbb{C}$) for full polarimetric interferometry as presented
in a companion paper, which generalizes and clarifies earlier work. Our
findings are relevant to state of the art co-polar and full polarimetric very
long baseline interferometry measurements to determine features very near the
event horizons of blackholes at the centers of M87, Centaurus A, and the Milky
Way.

Gauge theories; Gauge theory techniques; Geometric & topological phases;
Geometrical & wave optics; Group theory; Imaging & optical processing;
Interferometry; Lattice gauge theory; Mathematical physics; Radio, microwave,
& sub-mm astronomy

††preprint: APS/123-QED

##  I Introduction

Interferometry, which began with Young’s double slit experiment, has blossomed
over the years into a technique now widely used in physics, biology and
astronomy. Examples include the use of astronomical interferometry in the
first determination of the double-lobed morphology of Cygnus A [1], the first
imaging of the event horizon of a black hole in the center of the M87 galaxy
[2, 3, 4, 5, 6, 7, 8] and inner jet structures around the black hole in
Centaurus A [9], determination of crystal structures [10, 11, 12, 13, 14, and
references therein], seismic imaging [15, and references therein], remote
sensing using radar and sonar [16, 17, and references therein].

In each of these applications, accurate measurement of both the amplitude and
phase of the coherence (represented as complex values) is critical for
success. Often, accurate measurements of the complex-valued correlations is
made difficult by corruptions introduced by the propagation medium, and non-
ideal behavior of the instrument as well as of the measurement process. For
example, in radio interferometry, the electromagnetic wavefronts are corrupted
by ionospheric and tropospheric turbulences at low and high frequencies,
respectively, as well as by corrupting factors in the array element responses.
In optical interferometry, the atmospheric turbulence and imperfections in the
telescope’s surface geometry tend to destroy the phase information.

These element-based effects can be calibrated if there is a standard object of
known morphology available [18, 19]. An alternative strategy is self-
calibration [20, 21], which is a self-consistent scheme which uses iteration
and feedback to successively refine the image starting with an initial model.
But this is not always possible at the desired level of accuracy when the
signal-to-noise ratio or the a priori knowledge of the calibrator object is
inadequate. Therefore, there is considerable interest in calibration-
independent quantities that are unaffected by these element-specific factors,
and are hence true observable properties of the object’s structure.

Such interferometric invariants are typically constructed using the product of
pairwise correlations measured on a closed loop of array elements, and we call
them closure invariants. In co-polar interferometry (correlations between
measurements of the same polarization at all the array elements) at radio and
optical wavelengths, they are widely known as closure phases and amplitudes
[18, 19, 22, and references therein]. The analog of closure invariants include
the triplet and quartet invariants in X-ray crystallography [10, 11, 12, 13,
14, and references therein], and Bargmann invariants in quantum mechanics
[23]. A detailed geometric insight into closure phase and its connections to
various disciplines is provided in [24].

In astronomical interferometry, closure quantities can be traced to [25, 26].
Since then, they have been invaluable tools for interferometry at optical [27,
28, 29, 30, 22] and radio frequencies [31, 20, 32, 33, and references
therein]. This is particularly true of high spatial resolution applications
using very long baseline interferometry (VLBI) at radio frequencies. A famous
example is the recent imaging of the event horizon of the supermassive black
hole at the center of the M87 galaxy [2, 3, 4, 5, 6, 7, 8]. Recently, new
approaches of probing the structure formation in the intergalactic medium
through application of closure phases on faint spectral line emission from the
early universe during the cosmic reionization epoch ($z\gtrsim 6$) are also
being explored [34, 35, 36].

In the context of co-polar correlations in astronomical interferometry, a
detailed mathematical approach to determining the number of generic closure
invariants (including closure phases and closure amplitudes), and implications
for the resulting signal-to-noise ratios was presented by [37]. Recently,
first steps in the extension of closure invariants (called “closure traces”)
to correlations of full polarimetric antenna measurements were taken in [38],
which relied on including antenna auto-correlations to derive an independent
set of closure traces involving four correlations.

In this and an accompanying paper [39] (hereafter papers I and II,
respectively), we establish a formalism using a combination of group-theoretic
and linear algebraic approaches that advances the previous work [37, 38]. In
contrast to [38], we do not rely on the use of auto-correlations (which in
radio astronomy tend to be susceptible to significant noise biases, besides
instabilities caused by radio frequency interference and instrumental
systematics), but can incorporate them if reliably measured, and derive a
complete and independent set of closure invariants using gauge theory. In this
paper, we employ the $\mathrm{GL}(1,\mathbb{C})$ gauge group111In general,
$\textrm{GL}(n,\mathbb{C})$ refers to the group of nonsingular $n\times n$
matrices under multiplication, so $\textrm{GL}(1,\mathbb{C})$ is the group of
nonzero complex numbers under multiplication. and its associated gauge freedom
to treat and derive the complete and independent set of closure invariants in
the co-polar case. The closure phases and closure amplitudes familiar in radio
interferometery emerge naturally from our formalism which treats triangular
loops as fundamental units, and thus unifies the prescription for obtaining
closure phases and closure amplitudes. In paper II, we build on the
foundations presented in this paper and provide a formalism using the
$\textrm{GL}(2,\mathbb{C})$ gauge group and its associated gauge freedoms for
deducing the complete and independent set of closure invariants in the general
case of full polarimetric interferometry. In both papers, we also confirm our
findings with a parallel and independent viewpoint using conventional linear
algebra along with numerical simulations. We also generalize the analysis to
provide a prescription for extracting all the independent closure invariants
in an $N$-element interferometer array. These closure invariants represent
true observables about the system under observation, which in the case of
interferometric imaging in astronomy corresponds to the true physical
properties of the target object’s morphology.

This paper is structured as follows. Section II lays out the co-polar
interferometry context within which we seek a full set of independent
invariants. In section III, we present the expected number of independent
closure invariants through a dimension count analysis. Section IV develops a
formalism that provides multiple but equivalent methods to produce a complete
and independent set of closure invariants, which can be readily identified
with closure phases and closure amplitudes familiar in radio interferometry.
These invariants are listed explicitly in the case of three, four, and $N$
array elements. Section V summarizes this work. In Appendix A, we describe a
scheme for reliably measuring auto-correlations as the coincidence limit of
cross-correlations if an extra short-spaced pair of elements is available. In
Appendix B, we place our work in relation to the widely used ‘self-
calibration’ technique, which implicitly uses closure quantities. Appendix C
describes the numerical scheme used to confirm our analytic results, and
Appendix D brings out how our approach clarifies some aspects of the choice of
invarints as it relates to the effects of noise and to object model
parameters. In Appendix E, we highlight the explicit connection between the
closure phase and Pancharatnam phase [40, 41] in optics. Motivated by the
radio astronomy application, we extend the Pancharatnam rule to include
amplitudes as well.

##  II Co-polar interferometry

Consider an interferometer array with $N$ elements labeled by the indices
$a,b=0,\ldots,N-1$. Each element, $a$, in the array measures the amplitude and
phase of the same polarization (co-polar, or scalar) state of electric field,
$e_{a}$ (represented by a complex number), incident on it, which is
stochastic. The true correlation of the stochastic fields between pairs of
array elements is, by definition, obtained by cross-multiplication and
averaging, $S_{ab}\coloneqq\left\langle e_{a}e_{b}^{\dagger}\right\rangle$,
where, $\dagger$ denotes the conjugate transpose operator, which reduces to
complex conjugation for $\textrm{GL}(1,\mathbb{C})$ matrices in the co-polar
interferometric case dealt in this paper.

The complex electric fields, $e_{a}$, may be subject to arbitrary gains
$G_{a}$ at each element, where, $G_{a}$ are non-zero complex numbers
representing both amplitude and phase distortion of the measured signal. In
ideal conditions, all the gains $G_{a}$ would be unity and the measured signal
at each baseline would accurately reflect the true signal correlation,
$S_{ab}$, of emission from the object. In fact, the measured correlation,
$C_{ab}$, which is a corrupted form of the true correlation, can be written as

$\displaystyle C_{ab}$ $\displaystyle=G_{a}\left\langle
e_{a}e_{b}^{\dagger}\right\rangle
G_{b}^{\dagger}=G_{a}\,S_{ab}\,G^{\dagger}_{b}$ (1)

because of local gain distortions at each element. Generally, the cross-
correlations are complex-valued with non-zero amplitudes. From the definition,
cross-correlations satisfy $C_{ab}=C_{ba}^{\dagger}$. The auto-correlations,
$A_{aa}\coloneqq C_{aa}$, are real and positive. Our objective is to construct
quantities which are immune to the gain distortions and actually reflect true
properties of the object’s morphology, rather than local conditions at the
individual elements of the interferometer.

We relate this problem to the ‘gauge’ theories of fundamental forces like
electromagnetism. The local gains $G_{a}$ are regarded as gauge
transformations and our objective is to isolate a maximal set of gauge
invariant quantities which we will call closure invariants. This term
encompasses both closure amplitudes and closure phases. Our treatment places
them on an equal footing by working with a $\textrm{GL}(1,\mathbb{C})$ gauge
group.

##  III Counting arguments

Let us first do a dimensional count to see how many closure invariants we
would expect to find. The number of measured cross-correlations is $N(N-1)/2$,
this being the number of baselines or element pairs, i.e., the number of non-
repeating combinations among $N$ elements taken two at a time. Since each
cross-correlation is a complex number we have $N(N-1)$ real numbers. If
$n_{\textrm{A}}$ auto-correlations (described by $n_{\textrm{A}}$ real
numbers) are also measured, we would have to add $n_{\textrm{A}}$ to the above
count. Without loss of generality, we can assume that $n_{\textrm{A}}$ is
either zero or one222In radio astronomy, measuring any more auto-correlations
does not give us new information due to the implicit assumption of spatial
stationarity.. In radio astronomy, auto-correlation measurements are
unreliable because they are dominated by non-astronomical systematics. Our
formalism for closure invariants works with or without auto-correlations.

The unknown element-based gains are $N$ complex numbers, but note that there
may be sets of gains $G_{a}$ which do not affect the correlations. That is,
they satisfy $G_{a}C_{ab}G_{b}^{\dagger}=C_{ab}$ for all $a,b$. This yields
$G_{a}G_{b}^{\dagger}=1$ and so $G_{a}G_{a}^{\dagger}G_{b}G_{b}^{\dagger}=1$.
If the correlations themselves are unchanged, then so will the triple products
of correlations. Thus, using
$G_{a}G_{b}^{\dagger}G_{b}G_{c}^{\dagger}G_{c}G_{a}^{\dagger}=1$ (assuming
$N\geq 3$, and distinct $a,b,c$) gives us $G_{c}G_{c}^{\dagger}=1$ for all
$c$. This means $G_{a}=e^{i\theta}$ for all $a$, which is simply an overall
phase factor that cancels out in Eq. (1). This gives us a count of $2N-1$ real
numbers in the unknown gains.

Assuming none of the measured correlations is redundant (they are all
independent of each other) and all of them are used, the number of closure
invariants (that are independent of any choice of the unknown gains) is the
difference

$\displaystyle N(N-1)+n_{\textrm{A}}-(2N-1)$
$\displaystyle=N^{2}-3N+1+n_{\textrm{A}}\,.$ (2)

Note that this equals the number of closure phases, $(N-1)(N-2)/2$, plus the
number of closure amplitudes, $N(N-3)/2+n_{\textrm{A}}$ [18, 37]. For example,
this equals 1 (one closure phase) or 5 (three closure phases and two closure
amplitudes) when $n_{\textrm{A}}=0$ for $N=3$ or $N=4$, respectively, and 2
(one closure phase and one closure amplitude) or 6 (three closure phases and
three closure amplitudes) when $n_{\textrm{A}}=1$ for $N=3$ and $N=4$,
respectively. As $N$ becomes large, the structural information about the
object that can be extracted using closure phases and closure amplitudes
asymptotically approaches that which can be extracted using correlations.

##  IV Formalism for closure invariants

From a gauge theory perspective, we can regard each of the $N$ elements
$a=0,\ldots,N-1$ as vertices in a graph. The gains that represent element-
based corruption factors are local variables and multiplication by the gain
factor results in a gauge transformation at each vertex. Each baseline is an
edge $ab$ or link carrying “connection” variables that are bilocal since they
are defined on the link connecting two vertices. These are the cross-
correlations. In the language of gauge theory, each triangle,
$\Delta_{(abc)}\equiv(a,b,c)$ is called an “elementary plaquette” and defines
a “Wilson loop”. The Wilson loop of any closed circuit can be determined from
those of the elementary triangular plaquettes. Our objective is to list a
complete and independent set of elementary Wilson loops. These are the closure
invariants of radio astronomy.

In a $U(1)$ gauge theory like electromagnetism, the gauge group would be
unitary and complex conjugation in Eq. (1) would be the same as inversion. One
can then form independent closure invariants from each elementary triangular
plaquette $abc$ by simply multiplying correlations around the triangle, as in
$\mathcal{B}_{abc}=C_{ab}C_{bc}C_{ca}$. ($\mathcal{B}_{abc}$ is called the
“bispectrum” in interferometry). However, our gauge group
$\textrm{GL}(1,\mathbb{C})$ is not unitary. Using
$\mathcal{G}=\\{G_{a}\\},a=0,\ldots,N-1$ to denote a general gauge
transformation at every vertex, $a$, on the graph, the bispectrum and other
higher order products (“polyspectra”) of a set of $C_{ab}$ around a closed
loop, $\Gamma$, transform as

$\displaystyle\mathcal{P}_{\Gamma}$
$\displaystyle\xmapsto{\mathcal{G}}|G_{a}|^{2}\,|G_{b}|^{2}\ldots\mathcal{P}_{\Gamma}\,.$
(3)

The phase of $\mathcal{P}_{\Gamma}$ is invariant under gauge transformations
$\mathcal{G}=\\{G_{a}\\},a=0,\ldots,N-1$ and this is familiarly known as the
closure phase. However, because the gauge group is not unitary, the amplitude
of $\mathcal{P}_{\Gamma}$ is not invariant as it still depends on the gain
amplitudes, $|G_{a}|$.

Similarly, one can define closure amplitudes for every even-edged loop. For
example, given the loop $\Gamma=abcda$, the quantity

$\displaystyle\mathcal{Q}_{\Gamma}$
$\displaystyle=C_{ab}\,C^{-1}_{bc}\,C_{cd}\,C^{-1}_{da}$ (4)

transforms as

$\displaystyle\mathcal{Q}_{\Gamma}$
$\displaystyle\xmapsto{\mathcal{G}}G_{a}G^{\dagger-1}_{a}\,G_{b}G^{\dagger-1}_{b}\,G_{c}G^{\dagger-1}_{c}\,G_{d}G^{\dagger-1}_{d}\mathcal{Q}_{\Gamma}.$
(5)

It follows, since $\left|G_{a}G^{\dagger-1}_{a}\right|=1$ that
$\left|\mathcal{Q}_{\Gamma}\right|$, the modulus of $\mathcal{Q}_{\Gamma}$, is
a closure invariant:

$\displaystyle\left|\mathcal{Q}_{\Gamma}\right|$
$\displaystyle\xmapsto{\mathcal{G}}\left|\mathcal{Q}_{\Gamma}\right|.$ (6)

However, the phase of $\mathcal{Q}$ is not an invariant because it still
depends on the phases of $G_{a}$ at the loop vertices.

The virtues of $\mathcal{P}_{\Gamma}$ and $\mathcal{Q}_{\Gamma}$ can be
combined in a single quantity $\mathcal{C}_{\Gamma}$ by defining a hat
operator, $\widehat{Z}=(Z^{\dagger})^{-1}$, on non-zero complex numbers. We
introduce the term covariants as the set of even number of products of
correlations around a closed loop, $\Gamma$, with even numbered terms “hatted”
starting with the second. For example, a covariant on a 4-vertex loop can be
written as
$\mathcal{C}_{\Gamma}=C_{ab}\widehat{C}_{bc}C_{cd}\widehat{C}_{da}$. Hence,
covariants transform as

$\displaystyle\mathcal{C}_{\Gamma}$
$\displaystyle\xmapsto{\mathcal{G}}G_{0}\,\mathcal{C}_{\Gamma}\,G_{0}^{-1}\,.$
(7)

Because the gauge group of $G_{a}$, $\textrm{GL}(1,\mathbb{C})$, is Abelian,

$\displaystyle\mathcal{C}_{\Gamma}$
$\displaystyle\xmapsto{\mathcal{G}}\mathcal{C}_{\Gamma}\,.$ (8)

Thus, the covariants are invariant333Note that a four-edged covariant
essentially resembles a cross-ratio, which is a well-known invariant in
projective geometry [42, 43, and references therein]. under the gauge
transformation effected by the element-based corruption, and form the closure
invariants we are seeking in co-polar interferometry. Some of the notation
above is deliberately kept general so that it transfers easily to the more
difficult non-Abelian case discussed in paper II.

###  IV.1 Triangular Plaquettes

The need to have an even number of correlations in the product around a closed
loop appears to impose the requirement of elementary quadrilaterals rather
than triangles. However, it is unclear beforehand which even-vertexed loops
will provide us with a complete set of independent invariants. We show below
that such a set can be obtained by considering variables on elementary
triangular plaquettes that are independent, rather than quadrilateral
plaquettes. The latter can be entirely derived from the former.

The independent triangles in an $N$-element interferometer can be obtained by
fixing a base vertex (for example, 0) and choosing all the triangles that
contain this vertex [18]. There are $N_{\Delta}=(N-1)(N-2)/2$ such independent
triangles. We construct elementary triangular plaquette variables pinned at
vertex $0$ that includes just three elements ($0,a,b$):
$\mathcal{A}_{ab}=C_{0a}\widehat{C}_{ab}C_{b0}$. This quantity is neither a
bispectrum nor a covariant, but acts as a building block that can be used to
construct all the closure invariants. These triangular variables are clearly
independent and complete as all closed loops can be decomposed into triangles
and each distinct triangle is independent. Around the closed loop pinned at
vertex 0 ($\Gamma_{0}$), they undergo gauge transformations with $G_{0}$ as

$\displaystyle\mathcal{A}_{ab}$
$\displaystyle\xmapsto{\mathcal{G}}\left|G_{0}\right|^{2}\mathcal{A}_{ab}\,.$
(9)

We term such quantities as advariants in this paper. A special advariant is
$\mathcal{A}_{0}=A_{00}$, the auto-correlation. The notation clarifies that
this auto-correlation advariant is a local variable based at $0$. Note that
advariants have an advantage that their gauge freedom is restricted to just a
scaling by a single $|G_{0}|^{2}$ in contrast to polyspectra (including
bispectrum) whose gauge transformations depend on $|G_{a}|$ of all the
vertices in the loop. Thus, $|G_{0}|^{2}$ is the only unknown quantity in the
advariants that has to be eliminated to arrive at invariant quantities.
However, the phase of a triangular advariant is equal to the phase of the
corresponding bispectrum, and thus gives the closure phase [25], which is an
invariant.

###  IV.2 Complete and Independent Set of Invariants

####  IV.2.1 Method 1

Consider all the $N_{\Delta}$ elementary triangular plaquettes pinned at
vertex $0$. Each of them gives us a complex number $\mathcal{A}_{ab}$. These
complex numbers scale as in Eq. (9). The real and imaginary parts of these
complex numbers give us $M=2N_{\Delta}$ real numbers,
$\\{\widetilde{I}_{m}\\},m=1,\ldots,M$, all of which transform by the unknown
scale, $|G_{0}|^{2}$, under gauge transformations. The ratios of these
$\widetilde{I}_{m}$ with respect to any of them ($\widetilde{I}_{0}$, for
example) or some symmetric combination
($[\sum_{m}\widetilde{I}_{m}^{2}]^{1/2}$, for example), will eliminate this
unknown scale, and thus give us $M-1$ real closure invariants,
$\\{\widetilde{I}_{k}/\widetilde{I}_{0}\\},k=2,\ldots,M$. Their number is
$N^{2}-3N+1$, which is in exact agreement with the expectation in Eq. (2).
This gives us a complete and independent set of closure invariants. Also,
there is no need for auto-correlations in this construction. If we were to add
$A_{00}$ as a measured quantity, we would gain a single real invariant because
$\mathcal{A}_{0}=A_{00}$ being real and positive, would only contribute one
more real number to $\\{\widetilde{I}_{m}\\}$. Note that the practical problem
of systematic effects affecting auto-correlations can be mitigated by the use
of cross-correlations between a pair of closely spaced elements in cases where
the angular size of the object is much smaller than the angular resolution
corresponding to the short spacing, and is described in Appendix A.

While this method of taking ratios completely solves the problem we sought to
address, namely, the determination of a complete and independent set of co-
polar interferometric invariants, we provide below an alternative formalism
that allows us to better visualize the closure phases and amplitudes, as they
are familiarly known in radio interferometry, in terms of combinations of the
invariants determined here.

####  IV.2.2 Method 2

The elementary covariant quantity defined earlier as the closed loop product
of an even number of correlations with alternate terms hatted starting with
the second (a minimum of 4 terms) can be equivalently defined as the
multiplication of an even number (minimum of two) advariants with every
alternate term hatted starting with the second. For example, consider two
triangles, $\Delta_{1}\equiv\Delta_{(0,1,2)}$ (the base triangle) and
$\Delta_{\ell}\equiv\Delta_{(0,a,b)}$ chosen from the set of $N_{\Delta}$
triangles, both pinned at vertex 0, and their reversed forms,
$\nabla_{1}\equiv\Delta_{(0,2,1)}$ and $\nabla_{\ell}\equiv\Delta_{(0,b,a)}$.
The covariant, which is an invariant, can now be expressed by pairing the
advariants on these triangles as

$\displaystyle\mathcal{I}_{\Delta_{1};\Delta_{\ell}}=\mathcal{C}_{\Delta_{1};\Delta_{\ell}}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{\ell}}$
(10)
$\displaystyle\textrm{or,}\quad\mathcal{I}_{\Delta_{1};0}=\mathcal{C}_{\Delta_{1};0}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}\,,$ (11)

which are in general complex-valued. In the co-polar case studied in this
paper, since covariants are indeed invariants, they will be used
interchangeably. However, the same is not true in the full polarimetric case
presented in paper II.

The following properties of covariants are noted:

  * •

$\mathcal{A}_{\Delta_{1}}$ scales as $|G_{0}|^{2}$, whereas
$\widehat{\mathcal{A}}_{\Delta_{\ell}}$, and $\widehat{\mathcal{A}}_{0}$ scale
as $|G_{0}|^{-2}$, thereby cancelling the unknown scale factor in Eqs. (10)
and (11), and making the covariant an invariant.

  * •

A covariant formed with $\Delta_{1}$ and $\Delta_{\ell}$ has the same
amplitude as that formed with $\Delta_{1}$ and $\nabla_{\ell}$. That is,
$|\mathcal{I}_{\Delta_{1};\Delta_{\ell}}|=|\mathcal{I}_{\Delta_{1};\nabla_{\ell}}|$,
and thus do not provide independent amplitude information. Similarly,
$|\mathcal{I}_{\Delta_{\ell};\Delta_{1}}|=|\mathcal{I}_{\Delta_{1};\Delta_{\ell}}|^{-1}$.

  * •

If $\phi_{\Delta_{1}}$ and $\phi_{\Delta_{\ell}}$ are the phases of the
advariants (closure phases), then the phases of
$\mathcal{I}_{\Delta_{1};\Delta_{\ell}}$ and
$\mathcal{I}_{\Delta_{1};\nabla_{\ell}}$ are
$\phi_{\Delta_{1}}+\phi_{\Delta_{\ell}}$ and
$\phi_{\Delta_{1}}-\phi_{\Delta_{\ell}}$, respectively.

  * •

Each invariant, $\mathcal{I}_{\Delta_{1};\Delta_{\ell}}$, is complex-valued
and thus contains two real-valued invariants, except when
$\Delta_{\ell}=\Delta_{1}$. In that case,
$|\mathcal{I}_{\Delta_{1};\Delta_{1}}|=1$, and therefore, contains only one
real invariant, namely, its phase which is twice the closure phase,
$2\phi_{\Delta_{1}}$.

Using the above properties, a set of covariants (invariants) can be
constructed by pairing all the independent triangular advariants,
$\mathcal{A}_{\Delta_{\ell}}$ with the advariant of the base triangle,
$\mathcal{A}_{\Delta_{1}}$, as
$\mathcal{I}_{\Delta_{1}\Delta_{\ell}}=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{\ell}},\ell=1,\ldots,N_{\Delta}$.
Each of these complex invariants yields two real invariants for $\ell\neq 1$
and only one real invariant for $\ell=1$ as noted above. Therefore, the total
number of real invariants is $2N_{\Delta}-1=N^{2}-3N+1$. The presence of an
auto-correlation measurement will increase this count to $N^{2}-3N+2$ due to
one additional real-valued amplitude invariant in
$\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}$. Therefore, this method
provides $N^{2}-3N+1+n_{\textrm{A}}$ real invariants, which is a complete and
independent set, and is consistent with the first method using ratios as well
as with the dimension count analysis in section III. We also verified these
results numerically as detailed in Appendix C.

###  IV.3 Closure Phases and Amplitudes

Using the second method, we can directly identify the closure phases and
amplitudes familiar in radio interferometry with the closure invariants
obtained here.

The independent closure phases are directly given by the phases of the
$N_{\Delta}$ independent triangular advariants. But they can be derived from
the covariants, $\mathcal{I}_{\Delta_{1}\Delta_{\ell}}$, as well, where
$\Delta_{\ell}$ is any triangle that shares vertex 0 with the reference
triangle, $\Delta_{1}$. The phase of $\mathcal{I}_{\Delta_{1}\Delta_{1}}$ is
$2\phi_{\Delta_{1}}$, which yields $\phi_{\Delta_{1}}$ to within an ambiguity
of $\pi$. This ambiguity can be addressed by using the phase of the advariant,
$\mathcal{A}_{\Delta_{1}}$, which gives $\phi_{\Delta_{1}}$ directly without
the aforementioned ambiguity. The phase of the rest of the covariants
$\mathcal{I}_{\Delta_{1}\Delta_{\ell}}$ is
$\phi_{\Delta_{1}}+\phi_{\Delta_{\ell}}$, which will directly yield
$\phi_{\Delta_{\ell}}$ via plain substitution of the previously determined
$\phi_{\Delta_{1}}$. This will yield $N_{\Delta}$ independent closure phases.
The same construct also yields closure amplitudes.

An inspection of $|\mathcal{I}_{\Delta_{1}\Delta_{\ell}}|$ shows that it is
indeed a closure amplitude,
$|C_{01}||C_{21}|^{-1}|C_{20}||C_{a0}|^{-1}|C_{ab}||C_{0b}|^{-1}$. In general,
closure amplitudes formed from a pair of triangular advariants involves six
such terms. However, when $\Delta_{\ell}$ shares an edge with $\Delta_{1}$
($a=2$, for example), then the contributions, $|C_{20}|$ and $|C_{a0}|^{-1}$,
associated with that edge cancel each other, and the closure amplitude reduces
to the familiar form containing four terms,
$|C_{01}||C_{21}|^{-1}|C_{2b}||C_{0b}|^{-1}$. Noting that
$|\mathcal{I}_{\Delta_{1}\Delta_{1}}|$ does not contribute an amplitude (it
always has unit amplitude), there are $N_{\Delta}-1=N(N-3)/2$ independent
closure amplitudes (from the $N_{\Delta}-1$ triangle pairs sharing vertex 0
with the base triangle $\Delta_{1}$). The auto-correlation measurement, if
present, adds one more independent closure amplitude through
$|\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}|=|C_{01}||C_{21}|^{-1}|C_{20}|A_{00}^{-1}$.
Therefore, there are $N_{\Delta}-1+n_{\textrm{A}}$ independent closure
amplitudes in general. It may also be noted that if an auto-correlation can be
reliably measured, we can obtain all the invariants just from triangles with
the auto-correlation advariant as the base advariant using
$\mathcal{I}_{0;\Delta_{l}}=\mathcal{A}_{0}\widehat{\mathcal{A}}_{\Delta_{1}}$,
without requiring quadrilaterals or 6-edged loops.

The set of closure amplitudes and phases contains a total of
$2N_{\Delta}-1+n_{\textrm{A}}=N^{2}-3N+1+n_{\textrm{A}}$ real invariants as
given in Eq. (2), which is complete and independent, and is in agreement with
all the earlier analyses. Our approach using advariants and covariants
(invariants) constructed only from these triangular plaquettes provides a
unifying approach that directly identifies the traditional invariants known as
closure phases and amplitudes in radio interferometry.

Then the question arises: which form of closure invariants is preferable? In
Appendix D, we consider this question from the viewpoint of correlations
within the chosen set of invariants and their overall contribution to the
determination of the likelihood of parameterized models. In brief, the choice
of invariants is immaterial in the determination of the likelihood function of
the model parameters as long as the set of invariants employed is complete.
However, certain choices of coordinates in which the invariants are
represented could lead to coordinate-induced correlations among the
invariants, and also to situations when one or more of the invariants become
ill-defined in those coordinates. A well-known example occurs when
representing real and imaginary parts of random variable in polar form, where
the amplitude and phase angle can become correlated in general and the phase
angle becomes ill-defined in low $S/N$ regimes even if the real and imaginary
parts are perfectly uncorrelated and well-defined. For these reasons, even if
the real and imaginary parts of the invariants in our approach may be
correlated, we still prefer this representation over the polar form consisting
of phases and amplitudes.

###  IV.4 Illustration with Examples

For illustration, we consider the familiar examples of 3- and 4-elements
arrays below.

####  IV.4.1 3-element Array

Consider an array with $N=3$ and one auto-correlation measured ($n_{A}=1$). We
have 1 and 6 real values in the auto- and cross-correlations, respectively,
totaling 7 real measurements. We have $2N-1=5$ unknown real-valued parameters
in the three complex-valued element gains. From dimension counting used in Eq.
(2), we expect to find two real-valued closure invariants. From these
measurements, we can form a triangular advariant, $\mathcal{A}_{\Delta_{1}}$,
corresponding to the triangle $\Delta_{1}\equiv\Delta_{(0,1,2)}$, and an auto-
correlation advariant, $\mathcal{A}_{0}$. The complex invariant

$\displaystyle\mathcal{I}_{\Delta_{1};0}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}=\frac{|C_{01}||C_{20}|}{|C_{12}|A_{00}}\,e^{i\phi_{\Delta_{1}}}$
(12)

gives both these real-valued invariants. The magnitude of this complex
quantity is like a closure amplitude (with one auto-correlation) and the phase
is the standard closure phase of three elements. These two quantities are
evidently independent.
$\mathcal{I}_{\Delta_{1};\Delta_{1}}=e^{2i\phi_{\Delta_{1}}}$ will give no new
invariants.

If auto-correlation is not measured ($n_{A}=0$), the number of real
measurements reduces by 1 to 6. Hence, we now expect only one invariant from
Eq (2). Using the only advariant available, $\mathcal{A}_{\Delta_{1}}$, we
find

$\displaystyle\mathcal{I}_{\Delta_{1};\Delta_{1}}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{1}}=e^{2i\phi_{\Delta_{1}}}\,$
(13)

contains that expected invariant, namely, its phase, which is twice the
standard closure phase associated with $\Delta_{1}$, and carries an ambiguity
of $\pi$. However, the closure phase, $\phi_{\Delta_{1}}$, can be
unambiguously obtained from the phase of $\mathcal{A}_{\Delta_{1}}$ directly.

####  IV.4.2 4-element Array

For an $N=4$ array, we have $N_{\Delta}=3$ independent triangles,
$\Delta_{1}\equiv\Delta_{(0,1,2)},\Delta_{2}\equiv\Delta_{(0,2,3)},\Delta_{3}\equiv\Delta_{(0,1,3)}$,
and hence, three complex advariants,
$\mathcal{A}_{\Delta_{1}},\mathcal{A}_{\Delta_{2}}$, and
$\mathcal{A}_{\Delta_{3}}$. If the auto-correlation $\mathcal{A}_{0}$ is
measured we have an additional real advariant. The dimension count analysis
predicts six invariants when the auto-correlation is measured and five when it
is not.

When auto-correlation is not measured ($n_{\textrm{A}}=0$), we combine the
reference triangle $\Delta_{1}$, first with itself, and then with the two
other independent ones, $\Delta_{2}$ and $\Delta_{3}$.

$\displaystyle\mathcal{I}_{\Delta_{1};\Delta_{1}}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{1}}=e^{2i\phi_{\Delta_{1}}}$
(14) $\displaystyle\mathcal{I}_{\Delta_{1};\Delta_{2}}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{2}}=\frac{|C_{01}||C_{23}|}{|C_{21}||C_{03}|}e^{i(\phi_{\Delta_{1}}+\phi_{\Delta_{2}})}$
(15) $\displaystyle\mathcal{I}_{\Delta_{1};\Delta_{3}}$
$\displaystyle=\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{3}}=\frac{|C_{20}||C_{31}|}{|C_{21}||C_{30}|}e^{i(\phi_{\Delta_{1}}+\phi_{\Delta_{3}})}\,.$
(16)

We see that there are five real-valued invariants in the equations above –
three independent closure phases and two independent closure amplitudes
familiar in radio interferometry for a 4-element co-polar interferometer
array.

When $n_{\textrm{A}}=1$, we can write down the following three complex
invariants – $\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}$,
$\mathcal{A}_{\Delta_{2}}\widehat{\mathcal{A}}_{0}$, and
$\mathcal{A}_{\Delta_{3}}\widehat{\mathcal{A}}_{0}$ – which gives us six real
invariants comprising of three closure phases and three closure amplitudes
(two as above, and one more due to auto-correlation). The invariant phase in
$\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{\Delta_{1}}$ is redundant due
to the presence of $\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}$, and
can therefore be ignored.

Thus, the closure phases and closure amplitudes emerge naturally in our
approach which treats independent triangles as fundamental. We break this up
into a single invariant associated with the reference triangle, $\Delta_{1}$,
and two for each of the rest. The auto-correlation measurement will provide
one additional closure amplitude.

###  IV.5 Quadrilateral Closure Amplitudes

We have so far emphasized the role of triangles in constructing advariants.
The complete set of closure invariants we generate include four-sided loops
(quadrilaterals) as well as six-sided loops. Traditionally, closure amplitudes
have been discussed by astronomers using quadrilaterals. In order to connect
with this discussion we ask: Can we obtain a complete, independent set with
only quadrilateral-based closure amplitudes? A clear methodology is provided
in [44] which involves placing the $N$ elements on a ring and considering two
non-overlapping pairs of elements where each pair is made of consecutive
elements. Here, we provide an alternate method based on our formalism.

One way to ensure independence is to ensure that each loop contains a unique
element pair not present in any other. To do this, we fix the element pair
$(0,1)$ and consider the quadrilateral made with two more elements $a$ and
$b$. We arrange for $1<a<b<N$ and choose the set (which we call $\mathbb{A}$)
of closure amplitudes, $|C_{01}||C_{1a}|^{-1}|C_{ab}||C_{b0}|^{-1}$. So, each
such choice of a pair $(a,b)$ with $a<b$ gives us an independent closure
amplitude. The number of closure amplitudes in the set $\mathbb{A}$ is
therefore the number of such $(a,b)$ pairs, which is $(N-2)(N-3)/2$. However,
we know from basic counting [18] and linear algebra [37] that the total number
of independent closure amplitudes is $N(N-3)/2$. We therefore need
$(N-3)[N-(N-2)]/2=N-3$ more independent closure amplitudes to complete the
desired list. We supply these in a set $\mathbb{B}$, by observing that element
$0$ is only paired with $b$ and hence never paired with element $2$ in the set
$\mathbb{A}$. We therefore add precisely $N-3$ more closure amplitudes by
including the pair, $(0,2)$, as $|C_{01}||C_{1b}|^{-1}|C_{b2}||C_{20}|^{-1}$,
which contains $C_{02}$ in addition to our base choice of $C_{01}$. The set
$\mathbb{B}$ is only indexed by $b$ which can take $N-3$ values, and its
members are clearly independent of each other, since each contains the element
pair $(1,a)$ which is unique to it within this set $\mathbb{A}$. Every member
of $\mathbb{B}$ is also independent of all those of set $\mathbb{A}$, since
none of those contains the pair $(0,2)$. This completes the explicit
construction of $N(N-3)/2$ independent quadrilateral closure amplitudes.
Although this construction has been described in a self-contained way, it was
arrived at by considering quadrilateral closure amplitudes as the product of
two advariants, sharing an edge.

The closure amplitudes from our approach and the ring-based approach in [44]
are essentially equivalent. However, a consequence of the fundamental
difference in the methodologies leads to one notable difference in the
invariants constituting the complete and independent set when an $N+1$-th
element (indexed by $a=N$) is added to the array. In the approach of [44], the
inclusion of a new member, $a=N$, in the ring will remove the earlier pairing
of $(N-1,0)$ because they are no longer adjacent and the corresponding second
set of consecutive pairs will also get removed. In their place, two new pairs
of consecutive indices, $(N-1,N)$ and $(N,0)$, and the corresponding sets of
second adjacent pairings will have to be included. In other words, some
members of the original set of closure amplitudes will be removed and replaced
with a larger but different set. A similar removal and replacement process
also occurs when an array element has to be excluded. In our approach, the
additional element yields $N-1$ triangles (and advariants) pinned at base
vertex 0 containing this element that were not present before. When paired
with the base triangle, we get $N-1$ new complex-valued invariants in addition
to the existing set. That is, the members of the original set are undisturbed
and $2N-2$ real-valued invariants are added, amounting to a total of
$N^{2}-N-1$ real invariants for the $N+1$-element array. This difference may
have practical consequences when certain array elements are flagged at
specific times or frequencies due to poor quality of data. Our approach will
be robust to such practical challenges.

Thus, we have provided multiple methods of explicitly listing the closure
invariants in an $N$-element interferometer array. These methods do not yield
the same set of invariants, but carry exactly equivalent information. Our
approach unifies the treatment of all co-polar closure invariants, including
the triangular closure phases and quadrilateral closure amplitudes familiar in
radio interferometry as seen earlier.

##  V Summary

In this paper, we have explicitly enumerated from co-polar interferometric
measurements a complete and independent set of closure invariants that are
immune to corruptions local to the elements of an interferometer array. The
solution relies on mathematical ideas borrowed from gauge theories of particle
physics and the geometric phase of quantum mechanics and optics. Specifically,
the co-polar interferometric invariants presented here are invariant under
gauge transformations by the element-based corruption factors which belong to
the Abelian gauge group, $\textrm{GL}(1,\mathbb{C})$.

Invariants such as closure phases and closure amplitudes have been in wide use
in many disciplines for decades. However, they relied on using closed
triangular and quadrilateral loops, respectively, and thus required different
treatments. The solution presented here relies on using only independent
triangular plaquettes, which form the simplest non-trivial loops and are well
determined, as our basis. The quantity defined on these triangles, called
advariants, can be combined with an advariant chosen on a reference triangle
to produce a complete set of independent invariants. The main result of this
work is that using this formalism we have unified the treatment of closure
invariants, which have historically required separate methods. Thus, we have
provided an altered and unified perspective which improves our global
understanding of interferometric closure invariants from a symmetry viewpoint.
The familiar closure phases and closure amplitudes emerge naturally from this
approach.

Radio astronomers are aware that auto-correlation measurements are unreliable
because they could be dominated by non-astrophysical systematic effects. Our
formalism works even without auto-correlations, but can naturally accommodate
them if they are reliably measured. We have also outlined a methodology for
using a cross-correlation on a short-spaced element pair in place of an auto-
correlation to increase the number of invariants by 1, without introducing
systematic errors associated with a direct single-element auto-correlation
measurement.

Our approach clarifies certain aspects related to the choice of the form of
closure invariants and their implications for their covariance properties and
their impact on the likelihood of model parameters. While the methods
presented here are specific to co-polar interferometry, the concepts serve as
a stepping stone and reveal their full power when applied to a discussion of
invariants in full polarimetric interferometry in paper II. We expect that
this solution will aid astronomers in processing data collected from VLBI
measurements and other radio interferometry experiments.

There is also an unexpected spinoff from this application which is described
in more detail in Appendix E. As explained there, there is a strong connection
between closure phases in astronomy and the Pancharatnam phase in optics and
quantum mechanics. The present application suggests how the Pancharatnam phase
rule can be generalized to include amplitudes as well as phases. This rule
goes beyond existing ideas in the physics literature and shows how an
astronomical application can enrich the physics which is being applied.

###### Acknowledgements.

We thank Supurna Sinha, Ron Ekers, and the anonymous reviewers for a careful
reading of the paper. J. S. acknowledges support by a grant from the Simons
Foundation (677895, R.G.).

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##  Appendix A Auto-correlations from Short-spaced Elements

We have seen that auto-correlations can be used to construct covariants, not
just advariants, from a triangle of elements such as in Eq. (12). This option
is usually unavailable because of systematic errors in measuring auto-
correlations. However, when the angular size of the object
($\theta_{\textrm{obj}}$) is small compared to the angular resolution
determined by the element spacing, we can recover this advantage, by using a
cross-correlation, $C_{00^{\prime}}$, when the elements $0$ and $0^{\prime}$
have a small separation ($D_{00^{\prime}}$), such that
$\theta_{\textrm{obj}}\ll\lambda/D_{00^{\prime}}$, where, $\lambda$ is the
wavelength of observation.

We assume that the gains of the these closely spaced elements are independent
of each other, while their correlation in $C_{00^{\prime}}$ is free from the
problems that a single-element auto-correlation poses. One example would be an
additional element, $0^{\prime}$, in close proximity to the base element 0.
Another possibility is to use two physically close but independent subarrays
(denoted by 0 and $0^{\prime})$ that are phased from a dense array that is
being used as a single element in a “phased array” mode. An example is the
Atacama Large Millimeter/submillimeter Array (ALMA) in the EHT observations of
M87 and Centaurus A [3, 9]. Although $0^{\prime}$ can be paired with all the
other existing elements, the true correlations, $S_{a0^{\prime}}$, will carry
no new information that is not redundant444The redundancy, however, may be
useful towards obtaining a calibration that is independent of knowledge of the
sky brightness distribution, and forms the basis of redundant-calibration
schemes in radio interferometry [45, 46, for example] with $S_{a0}$, excepting
$S_{00^{\prime}}$, which is a good approximation to $S_{00}$.

Thus, we have two closely spaced elements, 0 and $0^{\prime}$, which do not
resolve the object’s features but have independent element gains, $G_{0}$ and
$G_{0^{\prime}}$. This means that $S_{00^{\prime}}\approx
S_{00}=S_{0^{\prime}0^{\prime}}$. We can form the covariant,
$C_{00^{\prime}}\widehat{C}_{0^{\prime}a}C_{ab}\widehat{C}_{b0}=G_{0}S_{00^{\prime}}\widehat{S}_{0^{\prime}a}S_{ab}\widehat{S}_{b0}G_{0}^{-1}\approx
S_{00}\widehat{S}_{0a}S_{ab}\widehat{S}_{b0}$. Note that the $G_{0}$ terms are
eliminated because our group is Abelian and we have used
$S_{00^{\prime}}\approx S_{00}$, which is the true auto-correlation. Thus, we
have effectively included auto-correlations as the coincidence limit of cross-
correlations, which will increase the number of real-valued independent
invariants by 1 corresponding to the auto-correlation. The use of closely
located elements was suggested in [38] as a diagnostic for non element-
dependent errors. Here, we are using them to provide one more invariant
effectively involving an auto-correlation.

Since we have used advariants as the building blocks, we can also reformulate
this in terms of the advariant
$C_{00^{\prime}}\widehat{C}_{0^{\prime}a}C_{a0}=G_{0}S_{00^{\prime}}\widehat{S}_{0^{\prime}a}S_{a0}G_{0}^{\dagger}$,
where, $a\notin\\{0,0^{\prime}\\}$. Under our assumption of closeness of
elements 0 and $0^{\prime}$, this can be written as $\approx
G_{0}S_{00}\widehat{S}_{0a}S_{a0}G_{0}^{\dagger}$. Since
$\widehat{S}_{0a}S_{a0}=1$, we are therefore left with only
$G_{0}S_{00}G_{0}^{\dagger}$, which is exactly what an auto-correlation
advariant pinned at 0 would have yielded. This means the earlier discussion
with $n_{A}=1$ is applicable and we gain one invariant. This agrees with the
preceding discussion based on covariants, as expected.

We can also explain this using the dimension counting principles. The presence
of an additional $S_{00^{\prime}}\approx S_{00}$, which is nearly real and
positive, effectively increases the dimensionality of the real values in the
true correlations by $n_{\textrm{A}}=1$ to $N(N-1)+1$. It would appear that we
have introduced an unknown complex gain, $G_{0^{\prime}}$, consisting of two
real parameters in the process. However, because of the redundancy
$S_{a0^{\prime}}\approx S_{a0}$, $G_{0^{\prime}}$ can be expressed as

$\displaystyle G_{0^{\prime}}$ $\displaystyle\approx
C_{0^{\prime}a}C_{0a}^{-1}G_{0},$ (17)

which is fully determined by $G_{0}$, and therefore not an independent degree
of gauge freedom. Hence, the number of unknown real parameters in the gains is
still $2N-1$. Thus, the resulting number of real invariants increases by
$n_{\textrm{A}}=1$, to $N^{2}-3N+2$, which confirms the alternate viewpoints
presented above.

##  Appendix B Relation to self-calibration

The parameters describing the element-dependent effects can be determined if
there is a standard signal. In practice, measurements on a point-like object
are interspersed with those of the target object, to get a preliminary
calibration. A major improvement of this procedure is ‘self-calibration’ [20,
32, 21]. The approximate calibration parameters are only used as an initial
guess to produce an approximate image. The image is then refined by
alternating steps of deconvolution, with adjustment of the instrumental
parameters to best fit the current image at each stage. This converges, in
favourable cases, to a much better image than was earlier possible. The word
‘better’ reveals that criteria based on a priori information, such as
positivity, smoothness, and compactness of the emission, play a role via the
deconvolution step.

Our work is concerned with a related but distinct notion of using closure
invariants, which directly characterize the source and are independent of any
model or deconvolution scheme. Forming images using just these invariants
alone has been explored, especially in VLBI, by ‘forward-modeling’, that is
fitting a model to the measured closure invariants. This approach would be
appropriate in cases where the data are not extensive enough to constrain a
free-form fitting procedure like self-calibration. Invariants can be used to
discriminate between different proposed models, purely in the domain of
observations, without bringing in deconvolution with its attendant a priori
assumptions. A fully converged self-calibration solution will, of course,
automatically satisfy all invariants. However, invariants have a role even
when self-calibration is not directly applicable.

##  Appendix C Numerical Test for Independence of Invariants

Here, we numerically verify the independence of the invariants derived through
various analytical methodologies described in this paper. We begin with the
simplest case with three elements which gives three cross-correlations. We
also include one auto-correlation. We generate multiple realizations of one
random positive real number for the uncorrupted auto-correlation, $S_{00}$,
and three random complex numbers for the uncorrupted cross-correlations,
$S_{01}$, $S_{12}$, and $S_{31}$, and three more random complex numbers for
the element-based gains (corruptions), $G_{0}$, $G_{1}$, and $G_{2}$. These
can be used to construct the measured correlations, $A_{00}$, $C_{01}$,
$C_{12}$, and $C_{20}$ using Eq. (1), which are described by 7 real values.
From these, we construct the advariants, $\mathcal{A}_{\Delta_{1}}$ and
$\mathcal{A}_{0}$ and compute the triangular invariant,
$\mathcal{A}_{\Delta_{1}}\widehat{\mathcal{A}}_{0}$, whose real and imaginary
parts are the two closure invariants.

Analytically, the test for independence of these invariants would be to look
at the two real invariants as functions of the real and imaginary parts of the
input true correlations. We construct the Jacobian matrix of partial
derivatives relating first order changes in the output to those in the input.
The rank of the Jacobian matrix gives the number of independent invariants.
Numerically, we construct the elements of this Jacobian matrix by varying one
input quantity $x_{i}$ at a time by a small amount, and recording the output
changes $y_{j}$ as a column. In this case, there will be 7 columns of length
2, so the Jacobian is a $2\times 7$ matrix. The rank is checked numerically by
carrying out a singular value decomposition (SVD) of the Jacobian and
examining the list of non-zero singular values. Zero-valued singular values
appear as very small values (compared to the non-zero singular values by many
orders of magnitude) due to finite-precision computations. The number of zero
singular values (also called the corank) gives the reduction in the number of
independent invariants, compared to the total number being calculated. In
geometric terms, we are finding the dimension of the surface onto which a
given, general set of measured visibilities gets mapped when we compute a
number of invariants of our own choosing. This dimension is in general less
than the maximum possible rank, by the number of zeros.

The rank of a Jacobian can also be used to cross-check the dimension counting.
While we have pointed out the most obvious redundancy in the gains, namely, an
overall phase, one might want to verify that no others have been missed. In
this case, we simulate random true correlations as the input, and apply
randomly chosen gains to them to get the measured correlations as output. Now,
we vary all the gain parameters in small increments and find the Jacobian via
the partial derivatives of all the real-valued parameters in the measured
correlation with respect to these changes in the $2N$ real gain parameters.
Not surprisingly, in all cases the corank is 1, and the count of $2N-1$ for
the number of independent variations of the gains which modify the measured
correlations, is confirmed. This numerical scheme generalizes to full
polarimetric measurements as well.

##  Appendix D Choice of Invariants, Noise, and Imaging

We have seen in section IV that a complete and independent set of invariants
is not necessarily unique and can take multiple forms. How do we choose the
form for a complete and independent set of invariants from all possible sets?
We approach this question from two considerations – one from that of
likelihood on the inferred model parameter space, and another from anticipated
noise properties of the invariants themselves.

Let us consider determining the likelihood of a parametrized set of models,
e.g., a ring of emission with azimuthal asymmetries as in [7]. Here and below,
we use a single symbol like $\mathcal{M}$ to denote an entire set of
variables, in this case parameters describing the model. This model will lead
to a predicted set of correlations from which a predicted set of invariants
follows, $\mathcal{I}^{p}(\mathcal{M})$. The noise on the correlations,
presumed known, can be propagated to a probability distribution for the
invariants $\mathcal{I}$ around $\mathcal{I}^{p}(\mathcal{M})$, using the
functional relation $\mathcal{I}=f(C)$. We denote this density by
$P_{\mathcal{I}}(\mathcal{I})$. Evaluating this at the measured values of the
invariants $\mathcal{I}^{m}$ gives the likelihood function on the model space,
$L(\mathcal{M}|\mathcal{I}^{m})=P_{\mathcal{I}}(\mathcal{I}^{m})$.

Now consider working with two different, but complete, sets of invariants,
$\mathcal{I}$ and $\mathcal{\tilde{I}}$. There is a two-way functional
relation between them. Because the probability densities for $\mathcal{I}$ and
$\widetilde{\mathcal{I}}$ are related by
$P_{\mathcal{I}}(\mathcal{I})\,\mathrm{d}\mathcal{I}=P_{\widetilde{\mathcal{I}}}(\widetilde{\mathcal{I}})\,\mathrm{d}\widetilde{\mathcal{I}}$,
we get
$L(\mathcal{M}|\mathcal{I}^{m})=L(\mathcal{M}|\widetilde{\mathcal{I}}^{m})\,|\mathbf{J}|$,
where, $|\mathbf{J}|\coloneqq\det(\mathbf{J})$ is the determinant of the
Jacobian matrix, $\mathbf{J}\equiv
J_{pq}=\partial\widetilde{\mathcal{I}}_{p}/\partial\mathcal{I}_{q}$.
$|\mathbf{J}|$ is evaluated at the measured values of the two sets of
invariants, and does not depend on the model, $\mathcal{M}$. Hence,
$|\mathbf{J}|$ appears as a simple proportionality factor in the space of
$\mathcal{M}$. Therefore, the maximum likelihood solution does not depend on
the choice of invariants.

In Bayesian approaches, including the maximum entropy methods, there is an
additional factor, namely, a prior depending solely on $\mathcal{M}$. It may
be noted that the standard approach of self-calibration does not even use
invariants explicitly, while satisfying them implicitly, so the question of a
choice does not arise. It is therefore satisfying that direct use of
invariants in determining the maximum likelihood of the models is also
independent of the chosen form of invariants.

While the maximum likelihood of the models does not depend on the chosen form
of the invariants, there may however be other considerations of noise
characteristics and interpretations that may favour one form of invariants
over another. Here, we compare and contrast our approach to that of [44].
Their construction of $N(N-3)/2$ independent closure amplitudes, and the
demonstration that all other such amplitudes can be constructed from these is
elegant and complete, and has the convenience of interpretation in terms of
amplitudes and phases.

In our approach based on gauge theory, the use of triangles as the generators
of all other loops and their associated invariants is more natural. These
emerge as complex quantities, the magnitudes being generalized closure
amplitudes (since some have six baselines) and the phases are sums of the
standard closure phases on two triangles. We have chosen to work with the real
and imaginary parts rather than the amplitudes and phases of our set of
invariants. Our primary motivation is described below.

Following [18], for simplicity, consider an ideal case of bivariate Gaussian
distribution of two uncorrelated variables, $X$ and $Y$, denoting the real and
imaginary parts of a complex number, respectively. Their joint probability
distribution, without loss of generality, is taken to be centered on
$(X_{0},0)$ with a variance of $\sigma^{2}$, and is given by

$\displaystyle P_{X,Y}(X,Y)$
$\displaystyle=\frac{1}{2\pi\sigma^{2}}\exp{\left[-\frac{(X-X_{0})^{2}+Y^{2}}{2\sigma^{2}}\right]}\,.$
(18)

Here, $P_{X,Y}(X,Y)=P_{X}(X)P_{Y}(Y)$. By change of variables, $X=A\cos\theta$
and $Y=A\sin\theta$, the joint probability distribution in polar form becomes

$\displaystyle P_{A,\theta}(A,\theta)$
$\displaystyle=\frac{A^{\prime}}{2\pi\sigma^{2}}\exp{\left[-\frac{(A-X_{0})^{2}-2X_{0}^{2}\cos\theta+4(A-X_{0})X_{0}\sin^{2}\frac{\theta}{2}}{2\sigma^{2}}\right]}\,.$
(19)

where, $A^{\prime}=A\exp{\left[-X_{0}^{2}/\sigma^{2}\right]}$. The first and
the second terms inside the exponential in Eq. (19) depend only on $A-X_{0}$
and $\theta$, respectively. However, the third term depends on both, which
makes $A$ and $\theta$ correlated in general. Only when $S/N\gg 1$
($|A-X_{0}|\ll|X_{0}|$), the apparent correlation term may be neglected and
the joint distribution becomes separable into amplitude and phase terms,
$P_{A,\theta}(A,\theta)=P_{A}(A)P_{\theta}(\theta)$, where
$P_{\theta}(\theta)$ reduces to a von Mises distribution, that can be further
approximated as a Gaussian distribution in $\theta$ [18, 44]. However, when
$S/N\lesssim 1$, approximating that $A$ and $\theta$ are uncorrelated is not
only invalid, but the phase is also not well-defined. So, the joint
distribution is preferably represented in real and imaginary coordinates ($X$
and $Y$) rather than their polar form ($A$ and $\theta$). Thus, it is evident
that purely on account of the choice of the coordinate system on which the
joint probability distribution is represented, it can not only induce
covariance among the random variables but also cause one or more variables to
be poorly defined. Therefore, in such cases, a different coordinate system is
clearly preferred.

The closure invariants, regardless of the form they are represented in, are
generally higher order functions of the correlations, and are thus expected to
be neither Gaussian distributed nor have an uncorrelated behavior. When
expressed as amplitudes and phases, they could suffer from induced covariance
and singularities in low $S/N$ regimes. Thus, owing to such scenarios, we
consider the real and imaginary parts of the covariants as “better” variables
and prefer them over their amplitude and phase representation, even though
they carry the same physical information.

##  Appendix E Connection to Pancharatnam phases

The electric field time-series, $e_{a}(t)$, received at element, $a$, as a
function of time can be viewed as a vector in Hilbert space [47, 48],

$\displaystyle\ket{e_{a}}$ $\displaystyle=\\{e_{a}(t)\,|\,0\leq t\leq T\\}\,,$
(20)

where, $T$ is the integration time. The correlation function, $\left\langle
e_{a}e_{b}^{\dagger}\right\rangle$, provides an inner product between these
vectors,

$\displaystyle\braket{e_{b}}{e_{a}}$
$\displaystyle=C_{ab}=\frac{1}{T}\int_{0}^{T}e_{b}^{*}(t)\,e_{a}(t)\,\mathrm{d}t\,.$
(21)

Closure phases can now be understood as a Pancharatnam phase [40, 41] (see
[33] for an earlier radio astronomy application). Pancharatnam’s work in
polarization optics gives us a rule (known in mathematics as a “connection”)
for comparing phases between vectors based at $a$ and $b$. We fix
$\ket{e_{a}}$ at element $a$ and define $\ket{e_{b}}$ to be in phase with
$\ket{e_{a}}$ if the correlation $\braket{e_{b}}{e_{a}}$ is real and positive.
The physical motivation is that the intensity of the superposed beam achieves
a maximum under this condition. This gives a rule for transporting a phase
from element $a$ to $b$, that is, modifying the phase of $\ket{e_{b}}$ so that
it is “in agreement” with $\ket{e_{a}}$. Iterating this rule and going in a
closed loop from element $a$ to $b$ to $c$ and back to $a$, we find on
returning to $a$ that the cyclic application of the rule gives a non-trivial
phase change with respect to the original phase of $\ket{e_{a}}$. In words,
$\ket{e_{b}}$ can be “in phase” with $\ket{e_{a}}$, and $\ket{e_{c}}$ with
$\ket{e_{b}}$, but then $\ket{e_{c}}$ in general is not “in phase” with
$\ket{e_{a}}$. The Pancharatnam phase is

$\displaystyle\arg\braket{e_{b}}{e_{a}}\braket{e_{c}}{e_{b}}\braket{e_{a}}{e_{c}}$
$\displaystyle=\arg C_{ab}C_{bc}C_{ca}\,,$ (22)

which astronomers will recognize as the closure phase [25]. Closure phases,
thus, emerge as the curvature of the Pancharatnam rule for comparing phases.
This is a discrete version of the curvature familiar from parallel transport
along a closed curve on a sphere.

A natural question arises at this point: is there a transport rule for the
amplitude as well as the phase? The idea is to modify the vector $\ket{e_{b}}$
representing the signal at $b$ so that it is ‘in agreement’ with $\ket{e_{a}}$
in both phase and amplitude. This is achieved as follows. The vector
$\ket{e_{b}}^{\prime}=f_{b}\ket{e_{b}}$ is proportional to $\ket{e_{b}}$, but
is now rescaled by a complex factor $f_{b}$, i.e., in amplitude and phase so
that $\braket{e_{a}}{e_{b}^{\prime}}=f_{b}\braket{e_{a}}{e_{b}}=1$ (note that
in the case of phases this was only made real and positive, but the magnitude
was left undetermined). This gives $f_{b}=1/\braket{e_{a}}{e_{b}}$. At the
next stage, the vector $\ket{e_{c}}$ is rescaled to $f_{c}\ket{e_{c}}$ so that
its inner product with $f_{b}\ket{e_{b}}$ is 1, i.e.,
$f_{b}^{*}f_{c}\braket{e_{b}}{e_{c}}=1$. This gives
$f_{c}=\braket{e_{a}}{e_{b}}^{*}/\braket{e_{b}}{e_{c}}$. The pattern is now
clear. After an even number of steps, returning to $a$, we obtain a vector
$f_{a}\ket{e_{a}}$ which can be compared in amplitude and phase to the
original $\ket{e_{a}}$. The rescaling factor, $f_{a}$, comes out to be nothing
but our four element complex closure invariant, made up of correlations with
alternate terms hatted. This transport rule, applicable to amplitudes and
phases, can be regarded as a spinoff from radio astronomy to possible
application in other areas. The restriction to an even number of steps has
appeared before in relativity in a discussion of the analogue of Fermi
transport for null curves [49].