Original: Identifying Witnesses to Noise Transients in Ground-based Gravitational-wave Observations using Auxiliary Channels with Matrix and Tensor Factorization Techniques
Revised: Identifying Witnesses to Noise Transients in Ground-based Gravitational-wave Observations using Auxiliary Channels with Matrix and Tensor Factorization Techniques

Original: Abstract
Revised: Abstract

Original: Ground-based gravitational-wave (GW) detectors are a frontier large-scale experiment in experimental astrophysics.
Revised: Ground-based gravitational-wave (GW) detectors are a frontier large-scale experiment in experimental astrophysics.

Original: Given the elusive nature of GWs, the ground-based detectors have complex interacting systems made up of exquisitely sensitive instruments which makes them susceptible to terrestrial noise sources.
Revised: Given the elusive nature of GWs, the ground-based detectors have complex interacting systems made up of exquisitely sensitive instruments which makes them susceptible to terrestrial noise sources.

Original: As these noise transients - termed as glitches - appear in the detector’s main data channel, they can mask or mimic real GW signals resulting in false alarms in the detection pipelines.
Revised: As these noise transients - termed as glitches - appear in the detector’s main data channel, they can mask or mimic real GW signals resulting in false alarms in the detection pipelines.

Original: Given their high rate of occurrence compared to astrophysical signals, it is vital to examine these glitches and probe their origin in the detector’s environment and instruments in order to possibly eliminate them from the science data.
Revised: Given their high rate of occurrence compared to astrophysical signals, it is vital to examine these glitches and probe their origin in the detector’s environment and instruments in order to possibly eliminate them from the science data.

Original: In this paper we present a tensor factorization-based data mining approach, based on irregular tensor mining, to finding witness events to these glitches in the network of heterogeneous sensors that monitor the detectors and build a catalog which can aid human operators in diagnosing the sources of these noise transients.
Revised: In this paper we present a tensor factorization-based data mining approach, based on irregular tensor mining, to finding witness events to these glitches in the network of heterogeneous sensors that monitor the detectors and build a catalog which can aid human operators in diagnosing the sources of these noise transients.

Original:  Introduction
Revised:  Introduction

Original: Noise coupling into a large-scale, complex instrument’s main channel(s) is a typical issue in big scientific experiments that push the limits of technology towards ground-breaking discoveries.
Revised: Noise coupling into a large-scale, complex instrument’s main channel(s) is a typical issue in big scientific experiments that push the limits of technology towards ground-breaking discoveries.

Original: To tackle this issue, along with constant engineering improvements, these complex frontier experiments deploy a large array of monitoring systems for day-to-day diagnostics and calibration, producing vast quantities of raw data-streams about the overall state of the detectors.
Revised: To tackle this issue, along with constant engineering improvements, these complex frontier experiments deploy a large array of monitoring systems for day-to-day diagnostics and calibration, producing vast quantities of raw data-streams about the overall state of the detectors.

Original: This raw data, if strategically mined for latent patterns of interest, can potentially point to origins of noise couplings and aid the operators in their diagnostics.
Revised: This raw data, if strategically mined for latent patterns of interest, can potentially point to origins of noise couplings and aid the operators in their diagnostics.

Original: The raw data is often heterogeneous temporal data from which we need to engineer datasets suitable for data mining and the authors of this paper wish to explore the end-to-end matrix and tensor factorization based data mining pipeline in the context of complex scientific instruments like the ground-based gravitational wave detectors at LIGO.
Revised: The raw data is often heterogeneous temporal data from which we need to engineer datasets suitable for data mining and the authors of this paper wish to explore the end-to-end matrix and tensor factorization based data mining pipeline in the context of complex scientific instruments like the ground-based gravitational wave detectors at LIGO.

Original: The ground-based gravitational-wave (GW) detectors like advanced LIGO Aasi et al.
Revised: The ground-based gravitational-wave (GW) detectors like advanced LIGO Aasi et al.

Original: [2015] are one such frontier complex instrument that have successfully reached the state-of-the-art sensitivity needed to detect astrophysical signals.
Revised: [2015] are one such frontier complex instrument that have successfully reached the state-of-the-art sensitivity needed to detect astrophysical signals.

Original: Given their exquisite sensitivity, these detectors are plagued by various sources of transient terrestrial noise which affects the searches for GWs.
Revised: Given their exquisite sensitivity, these detectors are plagued by various sources of transient terrestrial noise which affects the searches for GWs.

Original: Glitches are non-Gaussian noise transients appearing in the main channel of the detector which measures the amount of strain 1 , h ( t ) , produced by a passing gravitational wave.
Revised: Glitches are non-Gaussian noise transients appearing in the main channel of the detector which measures the amount of strain 1 , h ( t ) , produced by a passing gravitational wave.

Original: Their origins are environmental and instrumental in nature.
Revised: Their origins are environmental and instrumental in nature.

Original: These noise transients stand out from the expected stationary Gaussian noise as transient power excesses which trigger GW search pipelines leading to false alarms and, along with the stationary noise, limit the detector’s sensitivity.
Revised: These noise transients stand out from the expected stationary Gaussian noise as transient power excesses which trigger GW search pipelines leading to false alarms and, along with the stationary noise, limit the detector’s sensitivity.

Original: Hence, glitch characterization is a crucial problem and has been studied at LIGO in two important ways - 1.
Revised: Hence, glitch characterization is a crucial problem and has been studied at LIGO in two important ways - 1.

Original: the time-frequency morphology of the glitches in the main channel, 2. presence of triggers in auxiliary diagnostic channels that are coincident with the glitches.
Revised: the time-frequency morphology of the glitches in the main channel, 2. presence of triggers in auxiliary diagnostic channels that are coincident with the glitches.

Original: Glitch’s morphology in the main channel Glitches have complex morphologies and vary in duration and frequency.
Revised: Glitch’s morphology in the main channel Glitches have complex morphologies and vary in duration and frequency.

Original: Therefore, LIGO scientists have been studying the glitches based on the morphological differences evident in their time-frequency spectrograms.
Revised: Therefore, LIGO scientists have been studying the glitches based on the morphological differences evident in their time-frequency spectrograms.

Original: Several different classes of glitches have been identified and given their diversity and abundance, a morphology-based glitch catalog, called the Gravity Spy catalog Zevin et al.
Revised: Several different classes of glitches have been identified and given their diversity and abundance, a morphology-based glitch catalog, called the Gravity Spy catalog Zevin et al.

Original: Bahaadini et al.
Revised: Bahaadini et al.

Original: Glanzer et al.
Revised: Glanzer et al.

Original: [2022], has been built using citizen scientists to label glitch images and machine-learning based image classification techniques.
Revised: [2022], has been built using citizen scientists to label glitch images and machine-learning based image classification techniques.

Original: Although this catalog of glitches based on their morphological characteristics has been meticulously built, it is important to find origins of these noise transients in the system and build a richer, more actionable catalog which can equip the operators to locate sources of these noise transients, understand coupling mechanisms and possibly eliminate these glitches with instrumental upgrades.
Revised: Although this catalog of glitches based on their morphological characteristics has been meticulously built, it is important to find origins of these noise transients in the system and build a richer, more actionable catalog which can equip the operators to locate sources of these noise transients, understand coupling mechanisms and possibly eliminate these glitches with instrumental upgrades.

Original: Auxiliary channels Apart from the main channel which measures the strain h
Revised: Auxiliary channels Apart from the main channel which measures the strain h

Original: ( t ) , LIGO maintains
Revised: ( t ) , LIGO maintains

Original: O (10 5 ) auxiliary channels to record the state of the instrument and its environment using a wide variety of sensors, some of which are used in calibration of the detector.
Revised: O (10 5 ) auxiliary channels to record the state of the instrument and its environment using a wide variety of sensors, some of which are used in calibration of the detector.

Original: Some of these channels may witness noise sources induce a glitch.
Revised: Some of these channels may witness noise sources induce a glitch.

Original: Finding correlations between glitches in the main channel and excess power events in the auxiliary channels is therefore useful in determining the origins of these glitches.
Revised: Finding correlations between glitches in the main channel and excess power events in the auxiliary channels is therefore useful in determining the origins of these glitches.

Original: Our contribution is a data mining approach for unsupervised analysis of glitches in the main channel and the triggers in the auxiliary diagnostic channels using matrix and tensor factorization .
Revised: Our contribution is a data mining approach for unsupervised analysis of glitches in the main channel and the triggers in the auxiliary diagnostic channels using matrix and tensor factorization .

Original: We provide a brief background on LIGO operations in section 2 followed by some existing related work and motivate the prevalence of matrix and tensor factorization techniques for a variety of real-world applications.
Revised: We provide a brief background on LIGO operations in section 2 followed by some existing related work and motivate the prevalence of matrix and tensor factorization techniques for a variety of real-world applications.

Original: We describe the end-to-end data mining pipeline in Section 3 beginning with how we performed data collection in Section 3.1, how we did feature engineering in Section 3.2 to create datasets as described in Section 3.3.
Revised: We describe the end-to-end data mining pipeline in Section 3 beginning with how we performed data collection in Section 3.1, how we did feature engineering in Section 3.2 to create datasets as described in Section 3.3.

Original: Section 4 describes the analysis results and Sectionconcludes the paper with some future directions for this line of work.
Revised: Section 4 describes the analysis results and Sectionconcludes the paper with some future directions for this line of work.

Original:  Background
Revised:  Background

Original: Gravitational-wave Astronomy with LIGO LIGO detectors are state-of-the-art laser interferometers, which, in absence of a gravitational wave are designed to register no signal at the output photo-diode.
Revised: Gravitational-wave Astronomy with LIGO LIGO detectors are state-of-the-art laser interferometers, which, in absence of a gravitational wave are designed to register no signal at the output photo-diode.

Original: When a gravitational wave passes through the earth, a relative change in arm lengths is induced which is measured in terms of strain h ( t ) , which is the main data product of LIGO.
Revised: When a gravitational wave passes through the earth, a relative change in arm lengths is induced which is measured in terms of strain h ( t ) , which is the main data product of LIGO.

Original: Unfortunately, however, transient terrestrial disturbances originating in the environment or the instrument can also cause a brief power excess and result in a signal being registered at the output photo-diode.
Revised: Unfortunately, however, transient terrestrial disturbances originating in the environment or the instrument can also cause a brief power excess and result in a signal being registered at the output photo-diode.

Original: This coupling of noise into the main channel, h ( t ) , is, in essence, a glitch.
Revised: This coupling of noise into the main channel, h ( t ) , is, in essence, a glitch.

Original: LIGO data analysts use the auxiliary channels as veto generators to remove time segments contaminated with glitches and generate clean segments of h ( t ) that are then searched for GW signals.
Revised: LIGO data analysts use the auxiliary channels as veto generators to remove time segments contaminated with glitches and generate clean segments of h ( t ) that are then searched for GW signals.

Original: Thus, as described in
Revised: Thus, as described in

Original: Essick et al.
Revised: Essick et al.

Original: [2013] it is useful to determine important auxiliary channels which can act as veto generators .
Revised: [2013] it is useful to determine important auxiliary channels which can act as veto generators .

Original: Given the large number of auxiliary channels, only some of which may witness a glitch, it is conducive to use machine learning-based techniques to find correlation between the main and auxiliary channel power excesses in an attempt to determine what type of mechanical couplings produces a certain type of glitch.
Revised: Given the large number of auxiliary channels, only some of which may witness a glitch, it is conducive to use machine learning-based techniques to find correlation between the main and auxiliary channel power excesses in an attempt to determine what type of mechanical couplings produces a certain type of glitch.

Original: Previous Efforts at LIGO There have been several efforts at LIGO that use these auxiliary channels for glitch analysis.
Revised: Previous Efforts at LIGO There have been several efforts at LIGO that use these auxiliary channels for glitch analysis.

Original: Notably, iDQ Essick et al.
Revised: Notably, iDQ Essick et al.

Original: [2020] is a machine learning-based low-latency glitch prediction pipeline which uses auxiliary channels information to train a binary classifier to compute
Revised: [2020] is a machine learning-based low-latency glitch prediction pipeline which uses auxiliary channels information to train a binary classifier to compute

Original: ( https:// the probability of presence of glitches in the main channel.
Revised: ( https:// the probability of presence of glitches in the main channel.

Original: Similarly, in Cavaglia et al.
Revised: Similarly, in Cavaglia et al.

Original: [2018] the authors use auxiliary channel information and binary classification as a tool to study 2 sets of well-understood glitches from the first two observing runs of LIGO.
Revised: [2018] the authors use auxiliary channel information and binary classification as a tool to study 2 sets of well-understood glitches from the first two observing runs of LIGO.

Original: Data Mining using Matrix and Tensor Factorization Techniques There is a wealth of works that leverage matrix and tensor factorization especially for exploratory unsupervised data mining of multi-modal data in a wide and seemingly disparate number of real-world applications.
Revised: Data Mining using Matrix and Tensor Factorization Techniques There is a wealth of works that leverage matrix and tensor factorization especially for exploratory unsupervised data mining of multi-modal data in a wide and seemingly disparate number of real-world applications.

Original: Examples include brain network discovery Davidson et al.
Revised: Examples include brain network discovery Davidson et al.

Original: [2013] and Electronic Health Record mining and extraction of disease phenotypes Ho et al.
Revised: [2013] and Electronic Health Record mining and extraction of disease phenotypes Ho et al.

Original: The motivation behind the choice of using matrix and tensor factorization to do these types of analysis is the computational efficiency and the simple and interpretable nature of these techniques.
Revised: The motivation behind the choice of using matrix and tensor factorization to do these types of analysis is the computational efficiency and the simple and interpretable nature of these techniques.

Original:  Proposed Data Mining Pipeline
Revised:  Proposed Data Mining Pipeline

Original: The task at hand is to find co-clusters of glitch events and their witness channels given a set of glitch events ( G ) occurring in the main channel ( h ( t ) ) with respect to the set of auxiliary channels ( A ) selected for the analysis.
Revised: The task at hand is to find co-clusters of glitch events and their witness channels given a set of glitch events ( G ) occurring in the main channel ( h ( t ) ) with respect to the set of auxiliary channels ( A ) selected for the analysis.

Original: In this paper, we formulate this task as a soft co-clustering problem which is discussed in brief in Section 3.3.
Revised: In this paper, we formulate this task as a soft co-clustering problem which is discussed in brief in Section 3.3.

Original: Figure 1 shows a diagrammatic representation of the complete data mining pipeline.
Revised: Figure 1 shows a diagrammatic representation of the complete data mining pipeline.

Original: We begin by choosing an analysis time interval in which we collect a set of glitch events that occurred in the main channel ( G ) and corresponding loud triggers that occurred in a set of auxiliary channels ( A ) around the time of the glitch events.
Revised: We begin by choosing an analysis time interval in which we collect a set of glitch events that occurred in the main channel ( G ) and corresponding loud triggers that occurred in a set of auxiliary channels ( A ) around the time of the glitch events.

Original: We represent this information first as a 2-mode tensor, i.e a matrix, where each element represents the occurrence of a trigger in an auxiliary channel for each glitch event.
Revised: We represent this information first as a 2-mode tensor, i.e a matrix, where each element represents the occurrence of a trigger in an auxiliary channel for each glitch event.

Original: We can later expand the matrix into a tensor to featurize the matrix elements with a set of features ( F ) related to the channels or the triggers in the channels.
Revised: We can later expand the matrix into a tensor to featurize the matrix elements with a set of features ( F ) related to the channels or the triggers in the channels.

Original: The constructed tensor is then factorized and the latent patterns resulting from the factors are examined using various validation tests that are specific to certain downstream applications.
Revised: The constructed tensor is then factorized and the latent patterns resulting from the factors are examined using various validation tests that are specific to certain downstream applications.

Original: 3.1 Data Collection
Revised: 3.1 Data Collection

Original: A channel is typically a raw data-readout over time from a monitoring sensor (or some useful derived quantity).
Revised: A channel is typically a raw data-readout over time from a monitoring sensor (or some useful derived quantity).

Original: Each channel can be tagged with certain features (properties) like - 1. which subsystem in the complex system it belongs to (for example, a temperature sensor belonging to the physical environment monitoring subsystem or an inertial sensor measuring ground motion belonging to the internal seismic isolation subsystem), 2. where it is located in the observatory’s infrastructure.
Revised: Each channel can be tagged with certain features (properties) like - 1. which subsystem in the complex system it belongs to (for example, a temperature sensor belonging to the physical environment monitoring subsystem or an inertial sensor measuring ground motion belonging to the internal seismic isolation subsystem), 2. where it is located in the observatory’s infrastructure.

Original: A trigger refers to a transient power excess event in a channel - h ( t ) or any a ( t ) - detected by an event trigger generator (ETG) like Omicron Robinet [2016] which runs continually in real-time at LIGO during its operational runs.
Revised: A trigger refers to a transient power excess event in a channel - h ( t ) or any a ( t ) - detected by an event trigger generator (ETG) like Omicron Robinet [2016] which runs continually in real-time at LIGO during its operational runs.

Original: Thus, glitches are simply loud triggers in h ( t ) that are not astrophysical in origin.
Revised: Thus, glitches are simply loud triggers in h ( t ) that are not astrophysical in origin.

Original: Omicron infers certain features (properties) of the generated triggers viz.
Revised: Omicron infers certain features (properties) of the generated triggers viz.

Original: signal-to-noise ratio (SNR), peak frequency, bandwidth, amplitude, duration, phase, etc.
Revised: signal-to-noise ratio (SNR), peak frequency, bandwidth, amplitude, duration, phase, etc.

Original: We can do a systematic trigger collection inspired by Cavaglia et al.
Revised: We can do a systematic trigger collection inspired by Cavaglia et al.

Original: [2018] as follows:
Revised: [2018] as follows:

Original: Obtain a set, G , of loud triggers, i.e. trigger with peak SNR > 7.5, that occurred in h ( t ) between t start and t end .
Revised: Obtain a set, G , of loud triggers, i.e. trigger with peak SNR > 7.5, that occurred in h ( t ) between t start and t end .

Original: For all t g ∈ G , find triggers from a set, A , of safe auxiliary channels in a fixed window [ t g − γ, t g + γ ] around each t g where, t g ∈ G is the peak SNR time of the glitch trigger.
Revised: For all t g ∈ G , find triggers from a set, A , of safe auxiliary channels in a fixed window [ t g − γ, t g + γ ] around each t g where, t g ∈ G is the peak SNR time of the glitch trigger.

Original: γ is window parameter to center the glitch trigger around its peak SNR time.
Revised: γ is window parameter to center the glitch trigger around its peak SNR time.

Original: In this analysis γ ranges from 0.1 to 1 second.
Revised: In this analysis γ ranges from 0.1 to 1 second.

Original: If no trigger is present in the window, a default null trigger is stored for that channel corresponding to that t g .
Revised: If no trigger is present in the window, a default null trigger is stored for that channel corresponding to that t g .

Original: In case of multiple loud triggers in the window, we pick the loudest trigger.
Revised: In case of multiple loud triggers in the window, we pick the loudest trigger.

Original: 3.2 Feature Engineering and Dataset Construction
Revised: 3.2 Feature Engineering and Dataset Construction

Original: Matrix Construction:
Revised: Matrix Construction:

Original: The matrix X ∈ R | G |×| A | encodes the presence or absence of a loud trigger in each auxiliary channel for every glitch in G weighted by peak SNR.
Revised: The matrix X ∈ R | G |×| A | encodes the presence or absence of a loud trigger in each auxiliary channel for every glitch in G weighted by peak SNR.

Original: Each | A | -length row of X corresponds to a glitch and each element X ( i, j ) is the peak SNR of i th trigger in j th auxiliary channel if trigger present, zero otherwise.
Revised: Each | A | -length row of X corresponds to a glitch and each element X ( i, j ) is the peak SNR of i th trigger in j th auxiliary channel if trigger present, zero otherwise.

Original: Tensor Construction:
Revised: Tensor Construction:

Original: The matrix described in the previous section can be extended into a tensor 4 to additionally encode the features (properties) of the 2 entities in the matrix i.e. the events and/or the channels.
Revised: The matrix described in the previous section can be extended into a tensor 4 to additionally encode the features (properties) of the 2 entities in the matrix i.e. the events and/or the channels.

Original: Structurally, we can create 2 types of the tensor as follows -
Revised: Structurally, we can create 2 types of the tensor as follows -

Original: “Regular” 3-mode cuboid tensor: X ∈ R | G |×| A |×| P | where, | P | = number of features.
Revised: “Regular” 3-mode cuboid tensor: X ∈ R | G |×| A |×| P | where, | P | = number of features.

Original: Thus, each element X ( i, j, k ) is the peak SNR of i th trigger in j th auxiliary channel the has the k th feature (property) if trigger present, zero otherwise.
Revised: Thus, each element X ( i, j, k ) is the peak SNR of i th trigger in j th auxiliary channel the has the k th feature (property) if trigger present, zero otherwise.

Original: “Irregular” tensor / Collection of matrices: X i ∈ R | G |×|
Revised: “Irregular” tensor / Collection of matrices: X i ∈ R | G |×|

Original: A i | where, i is the i th feature (property), i = 1 , ..., | P | .
Revised: A i | where, i is the i th feature (property), i = 1 , ..., | P | .

Original: Thus, each element X i ( m, n ) is the peak SNR of m th trigger in n th auxiliary channel that has the i th feature (property) if trigger present, zero otherwise.
Revised: Thus, each element X i ( m, n ) is the peak SNR of m th trigger in n th auxiliary channel that has the i th feature (property) if trigger present, zero otherwise.

Original: 3.3 Co-clustering glitches, channels and their features
Revised: 3.3 Co-clustering glitches, channels and their features

Original: Consider the traditional clustering problem where, a data matrix D ∈ R m × n has m data-points (or observations) each of which has n features.
Revised: Consider the traditional clustering problem where, a data matrix D ∈ R m × n has m data-points (or observations) each of which has n features.

Original: Clustering methods, like the popular k-means algorithm, partition the data-points to discover k subsets called clusters such that data-points in a cluster are similar to each other and distinct from data-points in other clusters.
Revised: Clustering methods, like the popular k-means algorithm, partition the data-points to discover k subsets called clusters such that data-points in a cluster are similar to each other and distinct from data-points in other clusters.

Original: Co-clustering refers to simultaneous clustering of multiple modes of the data.
Revised: Co-clustering refers to simultaneous clustering of multiple modes of the data.

Original: In case of the 2-mode data (i.e. a data matrix) D described above, co-clustering partitions along the n rows as well as the m columns to find subsets.
Revised: In case of the 2-mode data (i.e. a data matrix) D described above, co-clustering partitions along the n rows as well as the m columns to find subsets.

Original: Co-clustering can be formulated as a multi-linear factorization as described in Papalexakis and Sidiropoulos [2011] such that D m × n ≈
Revised: Co-clustering can be formulated as a multi-linear factorization as described in Papalexakis and Sidiropoulos [2011] such that D m × n ≈

Original: [ R m × k C k × n ] where k is referred to as the rank of the factorization.
Revised: [ R m × k C k × n ] where k is referred to as the rank of the factorization.

Original: In this paper, to co-cluster the 2-mode data matrix, we use Non-negative Matrix of
Revised: In this paper, to co-cluster the 2-mode data matrix, we use Non-negative Matrix of

Original: Gravity Spy Zevin et al.
Revised: Gravity Spy Zevin et al.

Original: [2017] which only registers triggers with peak SNR ≥ 7.5 as glitches in their catalog.
Revised: [2017] which only registers triggers with peak SNR ≥ 7.5 as glitches in their catalog.

Original: Since we want to use the Gravity Spy glitch catalog in this work to validate our findings, we will set the same threshold for the analyses presented in this paper.
Revised: Since we want to use the Gravity Spy glitch catalog in this work to validate our findings, we will set the same threshold for the analyses presented in this paper.

Original: A tensor in this context is a multidimensional array.
Revised: A tensor in this context is a multidimensional array.

Original: Factorization (NMF).
Revised: Factorization (NMF).

Original: The choice of NMF is dictated by the non-negative values (the SNRs of the triggers) in X .
Revised: The choice of NMF is dictated by the non-negative values (the SNRs of the triggers) in X .

Original: Imposing the non-negativity constraint can potentially yield more interpretable factors that prove useful for further analysis.
Revised: Imposing the non-negativity constraint can potentially yield more interpretable factors that prove useful for further analysis.

Original: To co-
Revised: To co-

Original: cluster the 3-mode data tensors which add trigger or channel related features to the event-channel pairs, we will use variants of PARAFAC tensor factorization.
Revised: cluster the 3-mode data tensors which add trigger or channel related features to the event-channel pairs, we will use variants of PARAFAC tensor factorization.

Original: We describe these factorization methods formally in the following sections.
Revised: We describe these factorization methods formally in the following sections.

Original: 3.3.1 Non-negative Matrix Factorization (NMF)
Revised: 3.3.1 Non-negative Matrix Factorization (NMF)

Original: The formulation of NMF is stated as follows.
Revised: The formulation of NMF is stated as follows.

Original: Given X ∈ R | G |×| A | + and desired number of components (rank) k << min ( | G | , | A | ) , find E ∈ R | G |× k + and C ∈ R
Revised: Given X ∈ R | G |×| A | + and desired number of components (rank) k << min ( | G | , | A | ) , find E ∈ R | G |× k + and C ∈ R

Original: k ×| A | + such that M ≈ EC .
Revised: k ×| A | + such that M ≈ EC .

Original: In our case, each row of E and C T is a k -length latent space representation of triggers and channels respectively.
Revised: In our case, each row of E and C T is a k -length latent space representation of triggers and channels respectively.

Original: For computation of NMF, we used the implementation from Scikit-Learn library available for Python.
Revised: For computation of NMF, we used the implementation from Scikit-Learn library available for Python.

Original: 3.3.2 Non-negative CP/PARAFAC Tensor Factorization
Revised: 3.3.2 Non-negative CP/PARAFAC Tensor Factorization

Original: In order to incorporate features of the trigger events, we can factorize X ∈ R | G |×| A |×| P | using the
Revised: In order to incorporate features of the trigger events, we can factorize X ∈ R | G |×| A |×| P | using the

Original: CP/PARAFAC tensor factorization.
Revised: CP/PARAFAC tensor factorization.

Original: As depicted in figure 1, the CP/PARAFAC tensor factorization of X is expressed as a sum of outer products of k rank-1 tensors called components or factors.
Revised: As depicted in figure 1, the CP/PARAFAC tensor factorization of X is expressed as a sum of outer products of k rank-1 tensors called components or factors.

Original: These components can be arranged as 3 factor matrices viz.
Revised: These components can be arranged as 3 factor matrices viz.

Original: E ∈ R | G |× k
Revised: E ∈ R | G |× k

Original: + , C ∈ R |
Revised: + , C ∈ R |

Original: A |× k + , and F ∈ R | P |× k + corresponding to each mode of X , each with r columns.
Revised: A |× k + , and F ∈ R | P |× k + corresponding to each mode of X , each with r columns.

Original: Formally stated as, X ≈ k (cid:88) r
Revised: Formally stated as, X ≈ k (cid:88) r

Original: = e r ◦ c r ◦ f r , where e r , c r and f r are the r th column in factor matrices E , C and F respectively.
Revised: = e r ◦ c r ◦ f r , where e r , c r and f r are the r th column in factor matrices E , C and F respectively.

Original: The true rank k of the tensor is defined as the minimum number of rank-1 components required to exactly reconstruct the original tensor.
Revised: The true rank k of the tensor is defined as the minimum number of rank-1 components required to exactly reconstruct the original tensor.

Original: However, a low-rank approximation of the tensor is of interest to our analysis since it can capture latent patterns across the modes of the tensor.
Revised: However, a low-rank approximation of the tensor is of interest to our analysis since it can capture latent patterns across the modes of the tensor.

Original: In our case, the rows of factor matrices E , C and F hold the k -length latent space representations of the glitch events, channels and trigger features respectively.
Revised: In our case, the rows of factor matrices E , C and F hold the k -length latent space representations of the glitch events, channels and trigger features respectively.

Original: For the computation of the PARAFAC decomposition we used the implementation from Tensorly Kossaifi et al.
Revised: For the computation of the PARAFAC decomposition we used the implementation from Tensorly Kossaifi et al.

Original: [2019] which uses hierarchical alternating least squares method.
Revised: [2019] which uses hierarchical alternating least squares method.

Original: Additionally, we impose non-negativity constraint on the factors.
Revised: Additionally, we impose non-negativity constraint on the factors.

Original: 3.3.3 PARAFAC-2 Tensor Factorization
Revised: 3.3.3 PARAFAC-2 Tensor Factorization

Original: In order to incorporate features of channels, we can jointly factorize the collection of matrices
Revised: In order to incorporate features of channels, we can jointly factorize the collection of matrices

Original: X i using the PARAFAC-2 factorization.
Revised: X i using the PARAFAC-2 factorization.

Original: The choice of irregular tensor mining technique like PARAFAC2 is made instead of CP/PARAFAC for incorporating features of channels to avoid obtaining trivially orthogonal slices (since each channel belongs to exactly one subsystem and location) in the third mode of the typical 3-mode cuboid tensor.
Revised: The choice of irregular tensor mining technique like PARAFAC2 is made instead of CP/PARAFAC for incorporating features of channels to avoid obtaining trivially orthogonal slices (since each channel belongs to exactly one subsystem and location) in the third mode of the typical 3-mode cuboid tensor.

Original: As depicted in figure 1, the PARAFAC-2 factorization of a collection of matrices where a matrix X i ∈ R | G |×|
Revised: As depicted in figure 1, the PARAFAC-2 factorization of a collection of matrices where a matrix X i ∈ R | G |×|

Original: A i | is a frontal slice in an irregular 3-mode tensor.
Revised: A i | is a frontal slice in an irregular 3-mode tensor.

Original: A i is the set of auxiliary channels in the i th subsystem or at the i th location.
Revised: A i is the set of auxiliary channels in the i th subsystem or at the i th location.

Original: Formally stated as X i
Revised: Formally stated as X i

Original: ≈ C
Revised: ≈ C

Original: i F i E T where k is the rank of the decomposition, C i ∈ R | A i |× k is factor matrix corresponding to the channels belonging to the i th subsystem or at the i th location, E ∈ R | G |× k is the factor matrix corresponding to the glitch events that is shared across all i factor matrices corresponding to channels and
Revised: i F i E T where k is the rank of the decomposition, C i ∈ R | A i |× k is factor matrix corresponding to the channels belonging to the i th subsystem or at the i th location, E ∈ R | G |× k is the factor matrix corresponding to the glitch events that is shared across all i factor matrices corresponding to channels and

Original: finally F i ∈ R k × k is a diagonal matrix indicating which clusters of channels in a specific subsystem or at a specific location are co-clustered with a cluster of glitches.
Revised: finally F i ∈ R k × k is a diagonal matrix indicating which clusters of channels in a specific subsystem or at a specific location are co-clustered with a cluster of glitches.

Original:  Results
Revised:  Results

Original: For this analysis, we chose a 10-day time interval in the first half of the third observing run of LIGO (referred to as O3a by the LIGO community)
Revised: For this analysis, we chose a 10-day time interval in the first half of the third observing run of LIGO (referred to as O3a by the LIGO community)

Original: Tse et al.
Revised: Tse et al.

Original: There were approximatelyloud glitches at the LIGO Hanford observatory.
Revised: There were approximatelyloud glitches at the LIGO Hanford observatory.

Original: We used approximately 550 safe auxiliary channels obtained from a list maintained by the Detector Characterization group at LIGO.
Revised: We used approximately 550 safe auxiliary channels obtained from a list maintained by the Detector Characterization group at LIGO.

Original: We construct the data matrix X described in 3.3.1, the data tensor X frequency as described in 3.3.2 using a feature related to the trigger events i.e. their peak frequencies and finally two variants for the irregular data tensor described in, 3.3.3 using 2 sets of features related to the auxiliary channels viz.
Revised: We construct the data matrix X described in 3.3.1, the data tensor X frequency as described in 3.3.2 using a feature related to the trigger events i.e. their peak frequencies and finally two variants for the irregular data tensor described in, 3.3.3 using 2 sets of features related to the auxiliary channels viz.

Original: a set of discrete subsystems that the channels belong to: X subsystems , 2.
Revised: a set of discrete subsystems that the channels belong to: X subsystems , 2.

Original: a set of discrete locations where the sensors corresponding to the channels are located: X locations .
Revised: a set of discrete locations where the sensors corresponding to the channels are located: X locations .

Original: Clustering glitch events The | G | -length vectors in E are indicators of clusters of glitchs.
Revised: Clustering glitch events The | G | -length vectors in E are indicators of clusters of glitchs.

Original: Given some external ground-truth labeling for the glitch events, like the Gravity Spy catalog, we can examine whether we find homogeneous clusters for the different Gravity Spy classes w.r.t.
Revised: Given some external ground-truth labeling for the glitch events, like the Gravity Spy catalog, we can examine whether we find homogeneous clusters for the different Gravity Spy classes w.r.t.

Original: the auxiliary channels obtained via co-clustering.
Revised: the auxiliary channels obtained via co-clustering.

Original: More specifically, to quantify homogeneity of the clusters found in the factors w.r.t.
Revised: More specifically, to quantify homogeneity of the clusters found in the factors w.r.t.

Original: the Gravity Spy classes, we count the number of occurrences of glitches belonging to each Gravity Spy class in the top n values of each | G | -length vector in E and aggregate this quantity across all factors.
Revised: the Gravity Spy classes, we count the number of occurrences of glitches belonging to each Gravity Spy class in the top n values of each | G | -length vector in E and aggregate this quantity across all factors.

Original: For example, homogeneity of a co-clustering instance with rank k is defined as 1 k (cid:80) ki
Revised: For example, homogeneity of a co-clustering instance with rank k is defined as 1 k (cid:80) ki

Original: =1 1 N i where N i is the number of unique Gravity Spy classes associated with top n values of i th factor (column) in E .
Revised: =1 1 N i where N i is the number of unique Gravity Spy classes associated with top n values of i th factor (column) in E .

Original: Here, n is adaptively chosen to be 90% of the norm of the factor.
Revised: Here, n is adaptively chosen to be 90% of the norm of the factor.

Original: Thus if we find highly homogeneous clusters for the Gravity Spy classes this quantity is close to 1.
Revised: Thus if we find highly homogeneous clusters for the Gravity Spy classes this quantity is close to 1.

Original: To quantify how many Gravity Spy classes a factorization instance covers, we define coverage as the fraction of unique Gravity Spy classes represented across all factors (columns) of E over the total number of classes in the dataset.
Revised: To quantify how many Gravity Spy classes a factorization instance covers, we define coverage as the fraction of unique Gravity Spy classes represented across all factors (columns) of E over the total number of classes in the dataset.

Original: We expect coverage to approach 1 as we increase the factorization rank.
Revised: We expect coverage to approach 1 as we increase the factorization rank.

Original: Selecting veto-generator channels The | A | -length vectors in C are indicators of clusters of channels.
Revised: Selecting veto-generator channels The | A | -length vectors in C are indicators of clusters of channels.

Original: More specifically, each vector encodes the contribution of a subset of channels to the corresponding subset of glitches.
Revised: More specifically, each vector encodes the contribution of a subset of channels to the corresponding subset of glitches.

Original: We used these vectors to obtain a set of candidate channels S ⊂ A , to test as witnesses of occurrences of glitches.
Revised: We used these vectors to obtain a set of candidate channels S ⊂ A , to test as witnesses of occurrences of glitches.

Original: We pick the channel corresponding to the maximum magnitude value in the vector, i.e. the highest contribution, and add it to S .
Revised: We pick the channel corresponding to the maximum magnitude value in the vector, i.e. the highest contribution, and add it to S .

Original: For each channel s i ∈ S , if there is a glitch in h ( t ) and a corresponding trigger in s i in a window around the glitch, we say s i witnessed the glitch.
Revised: For each channel s i ∈ S , if there is a glitch in h ( t ) and a corresponding trigger in s i in a window around the glitch, we say s i witnessed the glitch.

Original: For each channel s i ∈ S , we count the number of glitches in h ( t ) that it witnessed to test whether s i can be used as a veto generator and remove segments of h ( t ) before searching for real GW signals and reduce the number of false alarms.
Revised: For each channel s i ∈ S , we count the number of glitches in h ( t ) that it witnessed to test whether s i can be used as a veto generator and remove segments of h ( t ) before searching for real GW signals and reduce the number of false alarms.

Original: Channels-based glitch catalog Using factor matrices E and C we can label each glitch event with the channels that witness it.
Revised: Channels-based glitch catalog Using factor matrices E and C we can label each glitch event with the channels that witness it.

Original: Since each row i in E is an embedding of i th glitch event ( e i ) in the R k latent space discovered by the rank- k factorization, we can choose factor j from the k available factors which has the highest magnitude value for e i and associate the channels in the j th factor of C with e i .
Revised: Since each row i in E is an embedding of i th glitch event ( e i ) in the R k latent space discovered by the rank- k factorization, we can choose factor j from the k available factors which has the highest magnitude value for e i and associate the channels in the j th factor of C with e i .

Original: 4.1 Analysis
Revised: 4.1 Analysis

Original: Figure 2 shows factors discovered by a rank-5 Non-negative CP/PARAFAC instance for the data tensor X frequency .
Revised: Figure 2 shows factors discovered by a rank-5 Non-negative CP/PARAFAC instance for the data tensor X frequency .

Original: We can observe that the two factors (out of 5) shown in the figure have high homogeneity and find clusters of glitch events belonging to class #

Revised: We can observe that the two factors (out of 5) shown in the figure have high homogeneity and find clusters of glitch events belonging to class #

Original: and we can find the diagnostic channels that strongly witnesses this cluster is channel #281 in the first factor and channels #367 and #369 in the second factor.
Revised: and we can find the diagnostic channels that strongly witnesses this cluster is channel #281 in the first factor and channels #367 and #369 in the second factor.

Original: We can also observe that the first factor finds higher frequency witness triggers compared to the second factor.
Revised: We can also observe that the second factor finds higher frequency witness triggers compared to the first factor.

Original: Testing homogeneity of glitch clusters Although we can examine individual factors to discover such patterns, as shown in figure 3, the overall homogeneity of factors discovered by NMF across various choices of factorization rank ( k ) and coincidence window sizes ( γ ) for the glitch events in our dataset with respect to the external ground-truth (Gravity Spy) labels is low.
Revised: Testing homogeneity of glitch clusters Although we can examine individual factors to discover such patterns, as shown in figure 3, the overall homogeneity of factors discovered by NMF across various choices of factorization rank ( k ) and coincidence window sizes ( γ ) for the glitch events in our dataset with respect to the external ground-truth (Gravity Spy) labels is low.

Original: This suggests that morphological similarity in the time-frequency spectrograms of glitches as they appear in the main channel does not necessarily translate to having a common set of witness channels for the wide variety of Gravity Spy classes.
Revised: This suggests that morphological similarity in the time-frequency spectrograms of glitches as they appear in the main channel does not necessarily translate to having a common set of witness channels for the wide variety of Gravity Spy classes.

Original: This finding matches the one in Gurav et al.
Revised: This finding matches the one in Gurav et al.

Original: [2020] made about a set of glitch events in a different analysis period and w.r.t.
Revised: [2020] made about a set of glitch events in a different analysis period and w.r.t.

Original: a superset of auxiliary channels than ours.
Revised: a superset of auxiliary channels than ours.

Original: Testing channels for veto generation A veto generator is a channel used by LIGO data analysts to iteratively remove segments of h ( t ) that are contaminated with glitches before searching for astrophysical signals in it.
Revised: Testing channels for veto generation A veto generator is a channel used by LIGO data analysts to iteratively remove segments of h ( t ) that are contaminated with glitches before searching for astrophysical signals in it.

Original: If a channel has coincident triggers with h ( t ) glitches, the data segment around the trigger time is removed from h ( t ) .
Revised: If a channel has coincident triggers with h ( t ) glitches, the data segment around the trigger time is removed from h ( t ) .

Original: Thus an ideal veto generator will only have triggers that coincide with h ( t ) glitch events (true positives) to avoid removing valuable science data segments that don’t have a coincident glitch (false positives).
Revised: Thus an ideal veto generator will only have triggers that coincide with h ( t ) glitch events (true positives) to avoid removing valuable science data segments that don’t have a coincident glitch (false positives).

Original: Figure 3 shows 2 candidate witness channels selected by an NMF instance.
Revised: Figure 3 shows 2 candidate witness channels selected by an NMF instance.

Original: Channel #281 can be a good veto generator as the true positive rate exceeds the false positive rate above an SNR threshold of 25.
Revised: Channel #281 can be a good veto generator as the true positive rate exceeds the false positive rate above an SNR threshold of 25.

Original: This is not true for channel #300 which is a noisy channel with high rate of spurious loud triggers.
Revised: This is not true for channel #300 which is a noisy channel with high rate of spurious loud triggers.

Original: increases with increasing factorization ranks as expected.
Revised: increases with increasing factorization ranks as expected.

Original: Building a channels-based glitch catalog Figure 4 show how we can build a glitch catalog based on witness channels.
Revised: Building a channels-based glitch catalog Figure 4 show how we can build a glitch catalog based on witness channels.

Original: A rank 5 NMF instance selected the same 2 witness channels (out of approx.
Revised: A rank 5 NMF instance selected the same 2 witness channels (out of approx.

Original: 550 available channels) for 2 different glitch events occurring in h ( t ) .
Revised: 550 available channels) for 2 different glitch events occurring in h ( t ) .

Original: We can see these two channels have loud triggers in them around the time of these glitches but the morphology of the loud triggers differ for each glitch event as evident in the spectrograms of the trigger signals.
Revised: We can see these two channels have loud triggers in them around the time of these glitches but the morphology of the loud triggers differ for each glitch event as evident in the spectrograms of the trigger signals.

Original: The data analysis pipeline described in this paper enables us to systematically label each glitch event with a small subset of channels that witness it and further examine the morphology of the triggers in the witness channels.
Revised: The data analysis pipeline described in this paper enables us to systematically label each glitch event with a small subset of channels that witness it and further examine the morphology of the triggers in the witness channels.

Original: Thus far the morphology of only the glitches in the main channel ( h ( t ) ) have been exhaustively studied and cataloged by Gravity Spy since doing so for hundreds of auxiliary diagnostic channels is not feasible.
Revised: Thus far the morphology of only the glitches in the main channel ( h ( t ) ) have been exhaustively studied and cataloged by Gravity Spy since doing so for hundreds of auxiliary diagnostic channels is not feasible.

Original: The proposed data analysis pipeline addresses this feasibility issue by systematically clustering the glitch events w.r.t.
Revised: The proposed data analysis pipeline addresses this feasibility issue by systematically clustering the glitch events w.r.t.

Original: their witness channels instead of resorting to a brute force search.
Revised: their witness channels instead of resorting to a brute force search.

Original:  Conclusions and Future Directions
Revised:  Conclusions and Future Directions

Original: In this paper, we present an end-to-end data mining pipeline using unsupervised matrix and tensor factorization techniques to build a more comprehensive, data-driven catalog of noise transient events occurring in large-scale, complex scientific instruments like the ground-based gravitational wave detectors at LIGO.
Revised: In this paper, we present an end-to-end data mining pipeline using unsupervised matrix and tensor factorization techniques to build a more comprehensive, data-driven catalog of noise transient events occurring in large-scale, complex scientific instruments like the ground-based gravitational wave detectors at LIGO.

Original: Such catalogs can aid operators to study sources of noise events, investigate mechanisms through which noise gets coupled into the detector’s highly sensitive main channel and ultimately push the detectors towards their design sensitivity to achieve the scientific goal of observing larger volumes of spacetime to detect gravitational waves emanating from merging black holes and neutron stars.
Revised: Such catalogs can aid operators to study sources of noise events, investigate mechanisms through which noise gets coupled into the detector’s highly sensitive main channel and ultimately push the detectors towards their design sensitivity to achieve the scientific goal of observing larger volumes of spacetime to detect gravitational waves emanating from merging black holes and neutron stars.

Original: The procedures described to build useful datasets using hundreds of raw data-streams from a complex instrument’s infrastructure and the unsupervised machine learning techniques used to systematically mine this data are fairly generic and can be generalized to any similarly large-scale complex scientific instruments that are becoming prevalent in modern experimental physics.
Revised: The procedures described to build useful datasets using hundreds of raw data-streams from a complex instrument’s infrastructure and the unsupervised machine learning techniques used to systematically mine this data are fairly generic and can be generalized to any similarly large-scale complex scientific instruments that are becoming prevalent in modern experimental physics.

Original: The authors of this paper will test the proposed methods to build comprehensive catalog of noise events occurring in LIGO data at scale and extend the utility of this catalog for discovering and modelling the mechanisms of noise couplings in order to subtract them from the science data.
Revised: The authors of this paper will test the proposed methods to build comprehensive catalog of noise events occurring in LIGO data at scale and extend the utility of this catalog for discovering and modelling the mechanisms of noise couplings in order to subtract them from the science data.

Original:  Acknowledgements
Revised:  Acknowledgements

Original: The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.
Revised: The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459.

Original: Research at UCR was supported by the National Science Foundation under EAGER grant no.
Revised: Research at UCR was supported by the National Science Foundation under EAGER grant no.

Original: PHY 2141072 and CAREER grant no. IIS 2046086, and by a UCR Regents Faculty Development Award.
Revised: PHY 2141072 and CAREER grant no. IIS 2046086, and by a UCR Regents Faculty Development Award.

Original: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding party.
Revised: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding party.