Matched-filtering Techniques

& Deep Neural Networks

—— Application for Gravitational-wave Astronomy

He Wang (王赫)

Institute of Theoretical Physics, CAS

Member of KAGRA collaboration

Apache MXNet Day, 10:01 AM PST on December 14\(^\text{th}\), 2020

Based on DOI: 10.1103/physrevd.101.104003

hewang@mail.bnu.edu.cn / hewang@itp.ac.cn

OVERVIEW

  • Gravitational-wave (GW) Astronomy
  • Challenges & Opportunities
  • Motivation
  • Matched-filtering Convolutional Neural Network (MFCNN)
  • Lessons Learned

Gravitational Wave

from binary black hole merges

  • 100 years from Einstein's prediction to the first LIGO-Virgo detection (GW150914) announcement in 2016.
  • General relativity: "Spacetime tells matter how to move, and matter tells spacetime how to curve."
  • Time-varying quadrupolar mass distributions lead to propagating ripples in spacetime: GWs.

Gravitational-wave Astronomy

LIGO Hanford (H1)

LIGO Livingston (L1)

KAGRA

  • Advanced LIGO observing since 2015 (two detectors in US), joined by Virgo (Italy) in 2017 and KAGRA (Japan) in Japan in 2020.
  • Best GW hunter in the range 10Hz-10kHz.

Credit: Science Burrito

Challenges in GW Data Analysis

  • Event GW150914
    • On September 14th, 2015: GWs from two ~30 solar mass black holes (BHs), merging ~400Mpc from Earth (z~0.1), crossed the two LIGO detectors displacing their test masses by a small fraction of the radius of a proton.
    • Measuring intervals must be smaller than 0.01 seconds.

Credit: LIGO

  • Noise in the detector
    • The actual data from the detector is shown in gray.
    • The noise is much louder (~100x) than the expected signals
      in red, green and blue. (BHs with spinning/non-spining and two neutron stars)
    • It's non-Gaussian and non-stationary that containing anomalous non-Gaussian transients, known as glitches.

Credit: LIGO

Challenges in GW Data Analysis

Noise power spectral density (one-sided)

  • Matched-filtering Technique
    • It is an optimal linear filter for weak signals buried in Gaussian and stationary noise \(n(t)\).
    • Works by correlating a known signal model \(h(t)\) (template) with the data.
    • Starting with data: \(d(t) = h(t) + n(t)\).
    • Defining the matched-filtering SNR \(\rho(t)\):

 

 

            where

  • The 4-D search parameter space in the first observation run covered by a template bank to circular binaries for which the spin of the systems is aligned (or antialigned) with the orbital angular momentum of the binary.
  • ~250,000 template waveforms are used (computationally expansive).

Phys.Rev.X 6 (2016) 4, 041015

The template that best matches GW150914 event

\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2

...

 

 

Proof-of-principle studies

Production search studies

Milestones

More related works, see Survey4GWML (https://iphysresearch.github.io/Survey4GWML/)

  • When machine & deep learning meets GW astronomy:
    • Covering more parameter-space (Interpolation)
    • Automatic generalization to new sources (Extrapolation)

    • Resilience to real non-Gaussian noise  (Robustness)

    • Acceleration of existing pipelines (Speed, <0.1ms)

 
 

Opportunities in GW Data Analysis

Phys.Rev.Lett 120 (2019) 14, 141103

Stimulated background noises

Attempts on Real LIGO Noise

A specific design of the architecture is needed.

 [as Timothy D. Gebhard et al. (2019)]

Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012)

Classification

Feature extraction

Convolutional neural network (ConvNet or CNN)

  • CNNs always works pretty good on stimulated noises.
  • However, when on real noises from LIGO, this approach does not work that well.
  • It's too sensitive against the background + hard to find GW events)

Attempts on Real LIGO Noise

  • CNNs always works pretty good on stimulated noises.
  • However, when on real noises from LIGO, this approach does not work that well.
  • It's too sensitive against the background + hard to find GW events)

A specific design of the architecture is needed.

 [as Timothy D. Gebhard et al. (2019)]

Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012)

Classification

Feature extraction

Convolutional neural network (ConvNet or CNN)

 MFCNN

 MFCNN

 MFCNN

Motivation

Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012)

Classification

Feature extraction

Convolutional neural network (ConvNet or CNN)

  • With the closely related concepts between the templates and kernels , we attempt to address a question of:

Matched-filtering (cross-correlation with the templates) can be regarded as a convolutional layer with a set of predefined kernels.

>> Is it matched-filtering ?
>> Wait, It can be matched-filtering!
  • In practice, we use matched filters as an essential component of feature extraction in the first part of the CNN for GW detection.

Matched-filtering CNNs (MFCNN)

\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)

(whitening)

where

Time domain

Frequency domain

(normalizing)

(matched-filtering)

\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)
\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix} \bar{d}(t) = d(t) * \bar{S}_n(t) \\ \bar{h}(t) = h(t) * \bar{S}_n(t) \end{matrix}\right.
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)
  • Transform matched-filtering method from frequency domain to time domain.
  • The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):

Matched-filtering CNNs (MFCNN)

\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)

(whitening)

where

Time domain

Frequency domain

(normalizing)

(matched-filtering)

\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)
\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix} \bar{d}(t) = d(t) * \bar{S}_n(t) \\ \bar{h}(t) = h(t) * \bar{S}_n(t) \end{matrix}\right.
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)
  • Transform matched-filtering method from frequency domain to time domain.
  • The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):
  • In the 1-D convolution (\(*\)) on Apache MXNet, given input data with shape [batch size, channel, length] :
output[n, i, :] = \sum^{channel}_{j=0} input[n,j,:] \ast weight[i,j,:]

FYI:       \(N_\ast = \lfloor(N-K+2P)/S\rfloor+1\)

（A schematic illustration for a unit of convolution layer)

Matched-filtering CNNs (MFCNN)

\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)

(whitening)

where

Time domain

Frequency domain

(normalizing)

(matched-filtering)

\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)
\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix} \bar{d}(t) = d(t) * \bar{S}_n(t) \\ \bar{h}(t) = h(t) * \bar{S}_n(t) \end{matrix}\right.
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)
  • Transform matched-filtering method from frequency domain to time domain.
  • The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):

Apache MXNet Framework

modulo-N circular convolution

Matched-filtering CNNs (MFCNN)

  • The structure of MFCNN:

Input

Output

Matched-filtering CNNs (MFCNN)

Input

  • The structure of MFCNN:
C_0 = \mathop{\arg\max}_{C}\rho[1,C,N] \,,\\ N_0 = \mathop{\arg\max}_{N} \,\langle d \mid h\rangle[1,C_0,N]
  • In the meanwhile, we can obtain the optimal time \(N_0\) (relative to the input) of feature response of matching by recording the location of the maxima value corresponding to the optimal template \(C_0\).

Output

Matched-filtering CNNs (MFCNN)

  • The structure of MFCNN:

Input

C_0 = \mathop{\arg\max}_{C}\rho[1,C,N] \,,\\ N_0 = \mathop{\arg\max}_{N} \,\langle d \mid h\rangle[1,C_0,N]
  • In the meanwhile, we can obtain the optimal time \(N_0\) (relative to the input) of feature response of matching by recording the location of the maxima value corresponding to the optimal template \(C_0\).

Output

import mxnet as mx
from mxnet import nd, gluon
from loguru import logger

def MFCNN(fs, T, C, ctx, template_block, margin, learning_rate=0.003):
    logger.success('Loading MFCNN network!')
    net = gluon.nn.Sequential()         
    with net.name_scope():
        net.add(MatchedFilteringLayer(mod=fs*T, fs=fs,
                                      template_H1=template_block[:,:1],
                                      template_L1=template_block[:,-1:]))
        net.add(CutHybridLayer(margin = margin))
        net.add(Conv2D(channels=16, kernel_size=(1, 3), activation='relu'))
        net.add(MaxPool2D(pool_size=(1, 4), strides=2))
        net.add(Conv2D(channels=32, kernel_size=(1, 3), activation='relu'))    
        net.add(MaxPool2D(pool_size=(1, 4), strides=2))
        net.add(Flatten())
        net.add(Dense(32))
        net.add(Activation('relu'))
        net.add(Dense(2))
	# Initialize parameters of all layers
    net.initialize(mx.init.Xavier(magnitude=2.24), ctx=ctx, force_reinit=True)
    return net

The available codes: https://gist.github.com/iphysresearch/a00009c1eede565090dbd29b18ae982c

Lessons Learned

input

GW Data from GWOSC: https://www.gw-openscience.org/

This slide: https://slides.com/iphysresearch/apachemxnet_mfcnn

  •  Some benefits from MF-CNN architecture for GW detection:
    • Simple configuration for GW data generation and almost no data pre-processing.
    • It works on a non-stationary background.
    • Easy parallel deployments, multiple detectors can benefit a lot from this design.
    • Efficient searching with a fixed window.
  • Dr. Chris Messenger (University of Glasgow):
    “For me, it seems completely obvious that all data analysis will be ML in 5-10 years".
  • The main understanding of the algorithms for DL community:
    • GW templates are used as likely known features for recognition.
    • Generalization of both matched-filtering and neural networks.
    • Linear filtering (i.e. matched-filtering) in signal processing can be rewritten as deep neural networks (i.e. CNNs).
    • Rethinking the structure of neural networks? (data flow vs. weight flow)

Lessons Learned

input

  •  Some benefits from MF-CNN architecture for GW detection:
    • Simple configuration for GW data generation and almost no data pre-processing.
    • It works on a non-stationary background.
    • Easy parallel deployments, multiple detectors can benefit a lot from this design.
    • Efficient searching with a fixed window.
  • Dr. Chris Messenger (University of Glasgow):
    “For me, it seems completely obvious that all data analysis will be ML in 5-10 years".
  • The main understanding of the algorithms for DL community:
    • GW templates are used as likely known features for recognition.
    • Generalization of both matched-filtering and neural networks.
    • Linear filtering (i.e. matched-filtering) in signal processing can be rewritten as deep neural networks (i.e. CNNs).
    • Rethinking the structure of neural networks? (data flow vs. weight flow)
for _ in range(num_of_audiences):
    print('Thank you for your attention!')
Profile: iphysresearch.github.io/-he.wang/
Twitter: twitter.com/@Herb_hewang

GW Data from GWOSC: https://www.gw-openscience.org/

This slide: https://slides.com/iphysresearch/apachemxnet_mfcnn

Matched-filtering Techniques & Deep Neural Networks

By He Wang

Matched-filtering Techniques & Deep Neural Networks

Apache MXNet Day, 10:01 AM PST on December 14th, 2020

  • 1,156

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