# Burst behavior due to quasimode excited by stimulated Brillouin scattering
in high-intensity laser-plasma interaction

Q. S. Feng  Institute of Applied Physics and Computational Mathematics,
Beijing, 100094, China  L. H. Cao<EMAIL_ADDRESS>Institute of Applied
Physics and Computational Mathematics, Beijing, 100094, China  HEDPS, Center
for Applied Physics and Technology, Peking University, Beijing 100871, China
Collaborative Innovation Center of IFSA (CICIFSA) , Shanghai Jiao Tong
University, Shanghai, 200240, China  Z. J. Liu  Institute of Applied Physics
and Computational Mathematics, Beijing, 100094, China  HEDPS, Center for
Applied Physics and Technology, Peking University, Beijing 100871, China  C.
Y. Zheng<EMAIL_ADDRESS>Institute of Applied Physics and
Computational Mathematics, Beijing, 100094, China  HEDPS, Center for Applied
Physics and Technology, Peking University, Beijing 100871, China
Collaborative Innovation Center of IFSA (CICIFSA) , Shanghai Jiao Tong
University, Shanghai, 200240, China  X. T. He  Institute of Applied Physics
and Computational Mathematics, Beijing, 100094, China  HEDPS, Center for
Applied Physics and Technology, Peking University, Beijing 100871, China
Collaborative Innovation Center of IFSA (CICIFSA) , Shanghai Jiao Tong
University, Shanghai, 200240, China

###### Abstract

The strong-coupling mode, called “quasimode”, will be excited by stimulated
Brillouin scattering (SBS) in high-intensity laser-plasma interaction. And SBS
of quasimode will compete with SBS of fast mode (or slow mode) in multi-ion
species plasmas, thus leading to a low-frequency burst behavior of SBS
reflectivity. The competition of quasimode and ion-acoustic wave (IAW) is an
important saturation mechanism of SBS in high-intensity laser-plasma
interaction. These results give a clear explanation to the low-frequency
periodic burst behavior of SBS and should be considered as a saturation
mechanism of SBS in high-intensity laser-plasma interaction.

††preprint:

Backward stimulated Brillouin scattering (SBS), a three-wave interaction
process where an incident electromagnetic wave (EMW) decays into a
backscattered EMW and a forward propagating ion-acoustic wave (IAW), leads to
a great energy loss of the incident laser and is detrimental in inertial
confinement fusion (ICF) He  _et al._ (2016); Glenzer  _et al._ (2010, 2007).
Therefore, SBS plays an important role in the successful ignition goal of ICF.
Multiple ion species are contained in the laser fusion program Neumayer  _et
al._ (2008). In the indirect-drive ICF Glenzer  _et al._ (2010, 2007) or the
hybrid-drive ignition He  _et al._ (2016), the inside of hohlraum will be
filled with low-Z plasmas, such as H or CH plasmas from the initial filled
material or from the ablated material off the capsule. Typically, in hybrid-
drive ICF, the laser intensity can reach as high as $I_{0}\sim
10^{16}W/cm^{2}$, the strong-coupling modes will be excited by SBS. The
stimulated scattering process in this regime is referred to as SBS in the
strong coupling regime or in the quasimode regime. Liu  _et al._ (1974);
Guzdar  _et al._ (1996) The quasimode or the strong-coupling mode refers to
the modified low-frequency mode involved in the three-wave process, while the
electrostatic mode refers to a natural mode of the system in the absence of
the pump wave. The burst behavior of SBS reflectivity is universal in high-
intensity laser-plasma interaction, however, the cause is not clear and
require the explanation. Besides, understanding the excitation of different
IAW modes and competition between SBS of different modes under the condition
of high-intensity laser plasma interaction is vital to predict SBS laser
energy losses and to improve the energy coupling into the fusion capsule.

Many mechanisms for the saturation of SBS have been proposed, including
frequency detuning due to particle trapping Froula  _et al._ (2002), coupling
with higher harmonics Cohen  _et al._ (1997); Rozmus  _et al._ (1992),
increasing linear Landau damping by kinetic ion heating Rambo  _et al._
(1997); Pawley  _et al._ (1982), the creation of cavitons in plasmas Weber
_et al._ (2005a, b) and so on. However, the burst behavior of SBS in high-
intensity laser-plasma interaction is confusing and has not been explained
well, which may be a potential saturation mechanism of SBS.

In this paper, we report the first demonstration that the strong-coupling mode
or quasimode will be excited by SBS and will coexist and compete with the IAW
in the high-intensity laser-plasma interaction. The competition between SBS of
IAW and SBS of quasimode will lead to the low-frequency burst behavior of SBS
reflectivity and decrease the total SBS reflectivity. Therefore the
competition of quasimode and IAW excited by SBS is an important saturation
mechanism of SBS in high-intensity laser-plasma interaction.

The wave number of IAW excited by backward SBS can be calculated by

$k_{A}\lambda_{De}\simeq 2\frac{v_{te}}{c}\sqrt{n_{c}/n_{e}-1},$ (1)

where $v_{te}=\sqrt{T_{e}/m_{e}}$ is the electron thermal velocity,
$n_{e},T_{e},m_{e}$ is the density, temperature and mass of the electron.
Considering fully ionized, neutral, unmagnetized plasmas with the same
temperature of all ion species ($T_{H}=T_{C}=T_{i}$), the linear dispersion
relation of the ion acoustic wave in multi-ion species plasmas is given by
Williams  _et al._ (1995); Feng  _et al._ (2016a, b)

$\epsilon(\omega,k=k_{A})=1+\chi_{e}+\sum_{\beta}\chi_{i\beta}=0,$ (2)

where $\chi_{e}$ is the susceptibility of electron and $\chi_{i\beta}$ is the
susceptibility of ion $\beta$.

When the strong pump light interacts with plasmas, the strong-coupling mode
will be generated, which will grow with time but not be damped and is called
quasimode. The dispersion relation of quasimode in strong pump light is given
by Drake  _et al._ (1974); Zhan-Jun  _et al._ (2012)

$\epsilon(\omega_{A},k_{A})=\frac{k_{A}^{2}v_{os}^{2}}{4}\chi_{e}(1+\sum_{\beta}\chi_{i\beta})(\frac{1}{D^{-}}+\frac{1}{D^{+}}),$
(3)

where $D^{-}\equiv D(\omega_{A}-\omega_{0},k_{A}-k_{0})$, $D^{+}\equiv
D(\omega_{A}+\omega_{0},k_{A}+k_{0})$, and
$D(\omega,k)=k^{2}c^{2}+\omega_{pe}^{2}-\omega^{2}$.

Under the condition of $T_{e}=5keV$, $n_{e}=0.3n_{c}$, one can obtain the wave
number of the IAW $k_{A}\lambda_{De}=0.3$ from Eq. (1). By solving Eq. (2) and
Eq. (3), the contours of solutions of the IAWs without pump light and with
strong pump light are shown in Fig. 1 under the condition of $T_{i}=0.2T_{e}$,
$k_{A}\lambda_{De}=0.3$. Although there are infinite solutions of the IAW as
shown in Fig. 1, the mode with the least Landau damping ($|Im(\omega)|$) will
be preferentially excited in SBS. There exist two groups of modes called “the
fast mode” and “the slow mode” in multi-ion species plasmas as shown in Fig.
1(a). Here, the fast mode and the slow mode refer to the least damped mode
belonging to each class of modes. When strong pump light interacts with
plasmas, the quasimode will be excited. As shown in Fig. 1(b), the imaginary
part of quasimode frequency is positive, which illustrates that the quasimode
is a growing mode.

Figure 1: (Color online) Contours of solutions to the dispersion relations of
(a) the fast IAW mode and the slow IAW mode without pump light, (b) the
quasimode with strong pump light $I_{0}=1\times 10^{16}W/cm^{2}$. Where the
red line is $Re[\epsilon]=0$ and the blue line is $Im[\epsilon]=0$. The
conditions are: $T_{e}=5keV$, $T_{i}=0.2T_{e}$, $n_{e}=0.3n_{c}$ and
$k_{A}\lambda_{De}=0.3$ in C2H plasmas.

An one dimension in space and one dimension in velocity (1D1V) Vlasov-Maxwell
code Liu  _et al._ (2009) is used to research the quasimode excited by SBS in
multi-ion species plasmas. We choose the high-temperature and high-density
region as an example, the electron temperature and electron density are
$T_{e}=5keV$, $n_{e}=0.3n_{c}$, where $n_{c}$ is the critical density for the
incident laser. The electron density is taken to be higher than $0.25n_{c}$,
thus the stimulated Raman scattering Feng  _et al._ (2018) and two-plasmon
decay instability Xiao  _et al._ (2015, 2016) are excluded. The C, C2H, CH, H
plasmas are taken as typical examples since they are common in ICF He  _et
al._ (2016); Glenzer  _et al._ (2007). The ion temperature is
$T_{i}=0.2T_{e}$. The linearly polarized laser intensity is $I_{0}=1\times
10^{16}W/cm^{2}$ with wavelength $\lambda_{0}=0.351\mu m$. The spatial scale
is [0, $L_{x}$] discretized with $N_{x}=5000$ spatial grid points and spatial
step $dx=0.2c/\omega_{0}$. And the spatial length is
$L_{x}=1000c/\omega_{0}\simeq 160\lambda_{0}$ with $2\times 5\%L_{x}$ vacuum
layers and $2\times 5\%L_{x}$ collision layers in the two sides of plasmas
boundaries. The plasmas located at the center with density scale length
$L=0.8L_{x}$ are collisionless. The incident laser propagates along the $x$
axis from the left to the right with outgoing boundary conditions. The strong
collision damping layers are added into the two sides of the plasmas
boundaries ($2\times 5\%L_{x}$) to damp the electrostatic waves such as IAWs
at the boundaries and decrease the effect of sheath field. The electron
velocity scale $[-0.8c,0.8c]$ and the ion velocity scale $[-0.03c,0.03c]$ are
discretized with $2N_{v}+1$ ($N_{v}=512$) grid points. The total simulation
time is $t_{end}=1\times 10^{5}\omega_{0}^{-1}$ discretized with
$N_{t}=5\times 10^{5}$ and time step $dt=0.2\omega_{0}^{-1}$.

Figure 2: (Color online) The frequency spectra of $Ey$ with the time scope $t\in[0,1\times 10^{5}]\omega_{0}^{-1}$ at $x_{0}=25c/\omega_{0}$. The parameters are $n_{e}=0.3n_{c},T_{e}=5keV,T_{i}=0.2T_{e},I_{0}=1\times 10^{16}W/cm^{2}$ in C2H plasmas as the same as Fig. 1(b).  Table 1:  The frequencies of different modes and corresponding scattering lights. The conditions are: $T_{e}=5keV$, $T_{i}=0.2T_{e}$, $n_{e}=0.3n_{c}$, $k_{A}\lambda_{De}=0.3$, and $I_{0}=1\times 10^{16}W/cm^{2}$ in C2H plasmas.  Mode | Theory | Simulation
---|---|---
$\omega_{A}/[10^{-3}\omega_{0}]$ | $\omega_{s}/\omega_{0}$ | $\omega_{s}/\omega_{0}$
fast mode | 2.7 | 0.9973 | 0.9972
slow mode | 1.9 | 0.9981 | $\setminus$
quasimode | 3.8 | 0.9962 | 0.9966

Figure 2 shows the spectra of the SBS scattering lights in the case of C2H
plasmas. Calculated by Vlasov simulation, the two peak frequencies of
scattering lights in C2H plasmas are
$\omega_{s}=0.9972\omega_{0},0.9966\omega_{0}$. Calculated from Eq. (2) and
Eq. (3), the frequency of the fast mode is $\omega_{f}=4.98\times
10^{-3}\omega_{pe}=2.73\times 10^{-3}\omega_{0}$, the frequency of the slow
mode is $\omega_{s}=3.49\times 10^{-3}\omega_{pe}=1.9\times 10^{-3}\omega_{0}$
and the frequency of the quasimode with pump light $I_{0}=1\times
10^{16}W/cm^{2}$ is $\omega_{q}=6.98\times 10^{-3}\omega_{pe}=3.82\times
10^{-3}\omega_{0}$. Therefore, the corresponding theoretical frequencies of
SBS scattering lights of the fast mode, the slow mode and the quasimode are
$\omega_{s}=0.9973\omega_{0},0.9981\omega_{0},0.9962\omega_{0}$. The
theoretical and simulation results are shown in Tab. 1. Compared to the
theoretical frequencies, we can see that the two peaks of the SBS scattering
lights in Fig. 2 are indeed the SBS of fast mode and SBS of quasimode.

Figure 3: (Color online) (a) The SBS reflectivities of different modes evolve
with time. Where SBS is the total SBS with the frequency scope
$\omega\in[0.9\omega_{0},0.999\omega_{0}]$, SBS of fast mode with
$\omega\in[0.9968\omega_{0},0.9977\omega_{0}]$ and SBS of quasimode with
$\omega\in[0.9960\omega_{0},0.9968\omega_{0}]$. (b) The reflectivity and the
transmitivity of the total SBS. The condition is as the same as Fig. 2.

Figure 3 gives the SBS reflectivities of fast mode and quasimode evolution
with time. As shown in Fig. 3(a), the SBS of fast mode will develop quickly
before $\sim 1\times 10^{4}\omega_{0}^{-1}$ and saturate during $t\in[1\times
10^{4},3\times 10^{4}]\omega_{0}^{-1}$. In this stage, the total SBS
reflectivity (blue line in Fig. 3(a)) is from the SBS of fast mode. After
$3\times 10^{4}\omega_{0}^{-1}$, SBS of quasimode will develop and compete
with SBS of fast mode, which will lead to the burst behavior of the total SBS
reflectivity. The SBS of fast mode demonstrates an obvious decrease when the
SBS of quasimode develops to the peak amplitude during $[3\times
10^{4},6\times 10^{4}]\omega_{0}^{-1}$. After $6\times 10^{4}\omega_{0}^{-1}$,
there is the contradictory trend between the SBS of fast mode and the SBS of
quasimode, which illustrates the competition of corresponding SBS of two
modes. The period of the total SBS reflectivity recurrence is
$T_{R}=7.501\times 10^{4}\omega_{0}^{-1}-6.494\times
10^{4}\omega_{0}^{-1}=1.007\times 10^{4}\omega_{0}^{-1}$ calculated from two
peaks of the SBS reflectivity as shown in Fig. 3(a), thus the frequecy of SBS
reflectivity recurrence is $\omega_{R}=1/T_{R}=9.93\times
10^{-5}\omega_{0}^{-1}$. In the same way, the period of SBS of fast mode is
$T_{f}=2.031\times 10^{4}\omega_{0}^{-1}$ and the period of SBS of quasimode
is $T_{q}=2.126\times 10^{4}\omega_{0}^{-1}$. Thus, the sum of two periods is
$\omega_{f}+\omega_{q}=1/T_{f}+1/T_{q}=9.63\times 10^{-5}\omega_{0}^{-1}$,
which is close to the period of SBS reflectivity recurrence $T_{R}$. This
illustrates that the burst behavior of SBS reflectivity is as a result of
competition between SBS of fast mode and SBS of quasimode, and the recurrence
period of SBS reflectivity burst comes from the sum of two periods of SBS of
fast mode and SBS of quasimode. Fig. 3(b) demonstrates the total reflectivity
and transmitivity of the SBS. Since the burst behavior of the total
reflectivity of SBS will occur due to the competition between SBS of fast mode
and SBS of quasimode, the transmitivity will demonstrate a same burst behavior
which is complementary to the SBS reflectivity after $3\times
10^{4}\omega_{0}^{-1}$. Therefore, the competition between SBS of fast mode
and SBS of quasimode is an important factor on the burst behavior of SBS and
the nonlinear saturation of SBS.

Figure 4: (Color online) The SBS reflectivities in different species plasmas
evolve with time.

Figure 4 demonstrates the SBS reflectivities in different species plasmas. In
stage I, the SBS reflectivities will develop and then saturate at a nearly
fixed level. In stage II, when different modes such as the fast IAW mode and
quasimode are excited in SBS in our simulation condition, other modes such as
slow IAW mode, ion bulk (IBk) mode Feng  _et al._ (2017a) could be excited in
SBS in some conditions, the competition among SBS of different modes will lead
to the low-frequency burst behavior of SBS reflectivity. With the rate of C to
H increasing, from Fig. 4(a) to Fig. 4(d), the Landau damping of IAW will
decrease and the SBS growth rate and SBS reflectivity will increase. As a
result, the beginning time of stage II will be earlier. However, for C
plasmas, the Landau damping of IAW is very low and the gain of SBS is very
high in the condition of $I_{0}=1\times 10^{16}W/cm^{2}$, the SBS cascade will
occur in stage III. Feng  _et al._ (2017b) The low-frequency burst in stage II
is due to the competition among SBS of different modes, while the high-
frequency burst in stage III is as a result of SBS cascade. These results in
Fig. 3 give a clear explanation of the low-frequency burst behavior in SBS,
which is common phenomena of SBS in high-intensity laser-plasma interaction.

Figure 5: (Color online) (a) The early linear stage of SBS in different
species plasmas. (b) The relation between the SBS reflectivity and SBS gain in
different species plasmas. Where the gains in multi-ion species plasmas, such
as CH and C2H plasmas, are calculated by the kinetic theory, and the gains in
single-ion species plasmas, such as H and C plasmas, are calculated by the
fluid theory. The SBS reflectivities by Vlasov simulation take the values at
$t=1.3\times 10^{4}\omega_{0}^{-1}$.

Figure 5(a) gives the linear growth and saturation process of SBS in the early
stage, in which the condition is as the same as Fig. 4. We can see that the
SBS in C plasmas increases the most quickly among the four cases, while the
growth rate of SBS in H plasmas is the slowest and the saturation level is the
lowest. With the rate of H to C in the plasmas increasing, the growth rate and
saturation level will decrease obviously, which is because of increase of the
IAW Landau damping. According to Fig. 5(a), the SBS reflectivity values at the
time of $t=1.3\times 10^{4}\omega_{0}^{-1}$ are chosen as the saturation
values, which is shown in Fig. 5(b), because they are stable at this time. The
linear gain of SBS by fluid theory is given by

$G=2\frac{\gamma_{0B}^{2}}{\nu_{A}v_{gs}}L,$ (4)

where

$\gamma_{0B}=\frac{1}{4}\sqrt{\frac{n_{e}}{n_{c}}}\frac{v_{0}}{v_{te}}\sqrt{\omega_{0}\omega_{A}}$
(5)

is the maximum temporal growth rate of SBS Berger  _et al._ (2015); Lindl  _et
al._ (2004), $v_{0}=eA_{0}/m_{e}c$ is the electron quiver velocity.
$v_{gs}=c^{2}k_{s}/\omega_{s}$ is the group velocity of SBS scattering light,
$L$ is the plasmas density scale length, $\omega_{A}\equiv Re(\omega_{A})$ and
$\nu_{A}\equiv|Im(\omega_{A})|$ are the frequency and Landau damping of IAW.
The collision damping of IAW can be neglected since the electron temperature
is as high as $T_{e}=5keV$ in our simulation, thus only the Landau damping of
IAW is considered. For single-ion species plasmas, such as H or C plasmas, the
gain of SBS can be calculated by the fluid theory. However, for multi-ion
species homogeneous plasmas, the SBS gain can be calculated by the kinetic
theory Lindl  _et al._ (2004):

$G(\omega_{s})=\frac{1}{4}\frac{k_{A}^{2}v_{0}^{2}L}{v_{gs}\omega_{s}}\text{Im}(\frac{\chi_{e}(1+\sum\limits_{i}\chi_{i})}{\epsilon(k_{s}-k_{0},\omega_{s}-\omega_{0})}),$
(6)

which is more precise than fluid theory. Where subscripts $0,s,A$ represent
the pump light, SBS scattering light and IAW. Under the strong damping
condition $\nu_{A}/\gamma_{0B}*\sqrt{v_{gs}/v_{gA}}\gg 1$ Forslund  _et al._
(1975), one can get the Tang model Tang (1966):

$R(1-R)=\varepsilon\\{\text{exp}[G(1-R)]-R\\},$ (7)

where $R$ is the reflectivity of SBS at the left boundary, and $\varepsilon$
is seed light at the right boundary. If $R\ll 1$, the Tang model can be
approximate to the seed amplification equation:

$R=\varepsilon\cdot\text{exp}(G).$ (8)

Although the strong damping condition in C plasmas or C2H plasmas is not
satisfied, the Tang model can also be applied to predicting the SBS
reflectivity in the linear saturation stage. As shown in Fig. 5(b), the SBS
reflectivities in different species plasmas are close to the Tang model. And
the seed amplification equation can not be applied to predicting the SBS
reflectivities in the condition of strong pump light, since the SBS
reflectivity $R$ is not much lower than 1. Figure 5 shows the linear process,
including linear growth and saturation process, in which the linear theory
such as Tang model is applicable. However, in the nonlinear process, such as
stage II and stage III in Fig. 4, the burst behavior will occur and the
nonlinear saturation mechanism of SBS has been explained in this paper.

In conclusions, quasimode will be excited by SBS in high-intensity laser-
plasma interaction, which will compete with the IAW excited by SBS. The
competition between SBS of quasimode and SBS of fast mode in C2H plasmas
demonstrates that this competition mechanism is the cause of the low-frequency
burst behavior of SBS. In different species plasmas, the same low-frequency
burst behavior will occur, which illustrates that the competition mechanism is
common in high-intensity laser-plasma interaction no matter what the plasmas
are. These results also give a good explanation of the intermediate low-
frequency burst and saturation process in C plasmas.

We would like to acknowledge useful discussions with C. Z. Xiao and L. Hao.
This research was supported by the National Natural Science Foundation of
China (Grant Grant Nos. 11875091, 11575035, 11475030 and 11435011), National
Postdoctoral Program for Innovative Talents (No. BX20180055), the China
Postdoctoral Science Foundation (Grant No. 2018M641274) and Science Challenge
Project, No. TZ2016005.

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End-to-end Learning for Early Classification of Time Series

Marc Rußwurmequal,tum
Sébastien Lefèvreubs
Nicolas Courtyirisa,ubs
Rémi Emonetuse
Marco Körnertum
Romain Tavenardequal,legt,irisa

tumTechnical University of Munich
ubsUniversité de Bretagne Sud
irisaInstitut de Recherche en Informatique et Systèmes Aléatoires
useUniversité Saint-Etienne
legtUniversité Rennes-LETG

Marc<EMAIL_ADDRESS>time series, early classification

Classification of time series is a topical issue in machine learning.
While accuracy stands for the most important evaluation criterion, some applications require decisions to be made as early as possible.
Optimization should then target a compromise between earliness, i.e., a capacity of providing a decision early in the sequence, and accuracy.
In this work, we propose a generic, end-to-end trainable framework for early classification of time series.
This framework embeds a learnable decision mechanism that can be plugged into a wide range of already existing models.
We present results obtained with deep neural networks on a diverse set of time series classification problems.
Our approach compares well to state-of-the-art competitors while being easily adaptable by any existing neural network topology that evaluates a hidden state at each time step.

§ INTRODUCTION

Qualitative sample of the learnable decision mechanism produced by a 1d convolutional model.
The model evaluates a hidden feature vector $\V{h}_t$ at each time step $t$ based on which classification confidences $\yhat_t$ and a stopping probability $\delta_t$ are derived.
To parameterize this stopping probability at training time, an attention-like score $P(t)$ is evaluated based on a given budget $\mathbb{B}_t$, as shown in the bottom two figures.

Classification of time series is a common problem in machine learning.
Learning methods, such as deep networks, are usually trained to solely reach the highest possible accuracy when predicting on a test set.
However, there are some contexts where the accuracy is not the single goal.
For instance, a doctor might want to make an accurate diagnosis early enough to treat the patient with the most appropriate medicine <cit.>.
Similarly, identifying droughts as early as possible by space-born imagery analysis allows local authorities to distribute resources efficiently and prevent famine and ultimately loss of life.
This discipline in machine learning is known as early classification <cit.>.

Early classification methods generally rely on i) learning a classifier per timestamp using a classification-only objective and ii) designing an early stopping criterion that utilizes predictions from these classifiers.

Conversely to existing works, we propose an end-to-end learning framework that is optimized jointly on accuracy and earliness by estimating a stopping probability $\delta_t$ based on a dual loss function.
To ensure that the stopping probability is parametrized accordingly with respect to our loss function, we design an attention-like mechanism $P(t) = \delta_t\mathcal{B}_t$.
This is used to focus the loss-penalty to specific times in the sequence and is subtracted from a monotonically decreasing attention budget $\mathcal{B}_t$.
A qualitative example in <ref>
illustrates these components.
Note that the stopping probability $\delta_t$ increases after processing a classification characteristic feature in the time series.
This indicates that enough classification-relevant information has been observed to make a decision at this time.

Our stopping probability is broadly applicable, since any differentiable model that estimates a hidden state $h_t$ at time instance $t$ can be modified with this mechanism.
We demonstrate this quantitatively and qualitatively on recurrent and convolutional model implementations.

§ RELATED WORK

In the following, we will address early classification of time series.
Time series are ordered sequences of observations of the form:

\begin{equation}
x = (x_0, x_1, \dots , x_T).
\end{equation}

We denote $\xuptot$ the partial observation of $x$ up to time $t$.
Early classification consists in predicting as early as possible the class of an incoming time series $x$.
In the literature, this translates as deciding, for any timestamp $t$, whether $\xuptot$ contains sufficient information to make a decision or if more data should be collected.
Hence, the early classification setting is related to the missing data problem.
Yet, it has its own peculiarities, due to the structure of the missing future data and the question whether more data should be waited for.

In a seminal paper, <cit.> suggest to delay decision-making for a nearest-neighbor classifier until neighborhood of the time series to be classified gets stable over time.
The method is named Early Classification on Time Series (ECTS).
This idea of delaying the decision until the classifier is reliable enough has been investigated in several other works.
<cit.> base their approach on an ensemble of classifiers for which a certain level of consensus has to be reached for the decision to be considered reliable.
RelClass <cit.> waits until it is highly probable that the decision made based on $\xuptot$ is the same as the decision that would be made based on the entire time series $x$.
Another example of this family of methods is EDSC <cit.> in which local features called shapelets <cit.> are extracted based on their predictive power.
A later feature selection scheme is then used to select those features that can predict both early and accurately.

One strong limitation of these methods is that they do not explicitly model the trade-off between earliness and accuracy.

To overcome this limitation, <cit.> proposed a framework for early classification which included the cost of delaying the decision in the loss function.
Clustering and cross-validation schemes are used to predict future classification costs.
From that information, a decision is made only if it is unlikely that the cost will decrease in the future.
elaborated on this method by formulating the decision to delay or not classification as a binary classification problem itself and removing the need for data clustering in the process.
In <cit.>, the authors also consider to explicitly optimize trade-off between earliness and accuracy.
This is done in a two-step process.
First, classifiers are learned based on a classification-only criterion.
Then, based on these classifiers, a heuristic parametric stopping rule is optimized using a genetic algorithm to minimize the target trade-off between earliness and accuracy.

In this work we will primarily compare accuracy and earliness by our method with <cit.>.
In order to easily compare performances on an equal footing, we will use in this paper a similar cost formulation

\begin{equation}
\mathcal{C} = \alpha \mathbb{I}_{\hat{\V{y}}_t \neq \V{y}} + (1-\alpha) \frac{t}{T} \, ,
\label{eq:cost_mori}
\end{equation}

where $t$ is the decision time and $\hat{\V{y}}_t$ is the classification decision itself.
This loss is the $\alpha$-weighted sum of a classification component and an earliness term $\frac{t}{T}$ as linear ratio relative to the sequence length $T$.
As they designed a non-differentiable stopping rule, they trained their method using a genetic algorithm.
With this framework, they opted for a straightforward indicator loss $\mathbb{I}_{\hat{\V{y}}_t \neq \V{y}}$ that counts the number of mis-classifications.
The trade-off parameter $\alpha$ is meant to be tuned based on the adopter's willingness to trade accuracy for an earlier classification.

We emphasize that these state-of-the-art methods require a separate classifier for each timestamp $t$.
In contrast, our stopping mechanism can be implemented on a single model that classifies time series data of arbitrary lengths.
Finally, in these methods, classifiers and stopping criteria are optimized sequentially, while we propose an approach in which a pre-trained model can be fine-tuned end-to-end using standard gradient back-propagation.

§ METHODOLOGY

In this section, we first describe our proposed differentiable decision-making mechanism that, after training, produces a stopping decision probability $\delta_t$.
The key feature of our approach is how we encourage the network to parameterize the weights $\theta_{\delta}$ of this stopping probability.
We start with designing a loss function that simultaneously optimizes for accuracy and earliness similar to <cit.>.
Based on $\delta_t$, we introduce a mechanism $P(t)$ that focuses the loss penalty on selected observations.

§.§ Mechanism

Let us assume we have a time series classification model that is able to compute, at each time step $t$, a hidden feature representation $h_t$
from which it derives class probability estimates $\yhat_t = \text{softmax}\left(\theta_\text{cl} {h}_t\right)$.
Given that classification probabilities are evaluated from this hidden representation $h_t$ at each time $t$, we hypothesize that the hidden state is likely to encode information about whether to stop at the current prediction or wait for more observations.

To utilize this information, we add a second linear output layer $\delta_t = \sigma\left(\theta_\delta {h}_t\right)$ with a sigmoidal non-linear activation function $\sigma(\cdot)$.
We interpret the output of this layer as the probability of stopping at time $t$.
At inference, we sample a hard stopping decision based on this probability.
A schematic illustration of the mechanism is shown in <ref>.

Schematic view of the elements at stake in our loss function. A temporal classification model produces classification probabilities $\yhat_t$ from a hidden state $h_t$ at each time $t$.
The model parameters are optimized via a classification loss function $\mathcal{L}_c$ and gradient back-propagation, as indicated by $\frac{\partial\mathcal{L}_{cl}}{\partial\theta_\text{cl}}$.
To implement a stopping probability $\delta_t$, we augment this model by an additional output $\delta_t = \sigma\left(\theta_\delta \V{h}_t\right)$.
The weights of this additional output $\theta_{\delta}$, however, can not be solely determined based on the classification loss $\mathcal{L}_c$ since no gradients can be back-propagated from $\mathcal{L}_c$ to $\theta_{\delta}$.
Hence, we multiply attention-like focus scores $P(t)$ to the evaluated loss to enable gradient propagation to $\theta_{\delta}$.

Given $\delta_t$, one can derive the probability of making a decision at time $t$

\begin{equation}
P(t) = \delta_t \cdot \underbrace{\prod_{\tau=0}^{t-1} 1 - \delta_\tau}_{\mathcal{B}_{t-1}} \, .
\label{eq:pts}
\end{equation}

In order to ensure that these $P(t)$ probabilities sum to one, we set $\delta_T$ to 1 regardless of $h_T$.

These decision probabilities $P(t)$ will be used to weight losses computed at all timestamps during training, such that decisions that are made at highly probable decision timestamps will induce larger gradients, as graphically illustrated in <ref>.

Since $P(t)$ is computed based on $\delta_t$ (as detailed later), gradients can now be back-propagated to $\theta_{\delta}$ and the stopping probability $\delta_t$ can be parametrized end-to-end.

With this mechanism, we see a certain analogy to attention scores <cit.> and, thus, propose a second interpretation.
If we interpret $P(t)$ as attention-like scores, one can see the product of all previous inverse probabilities as a budget $\mathcal{B}_{t-1}$ that is initialized with one.
At each time, $P(t) = \delta_t \cdot \mathcal{B}_{t-1}$ is calculated as a fraction of the remaining attention budget.
At the last timestamp $T$, we consume all the remaining budget $\mathcal{B}_{T-1}$ whatever the observations.

<ref> presents an example of the evolution of these quantities over time.

§.§ Loss function

Evaluated loss of common objective functions given the probability of the correct class $\yhat^+$.

Once equipped with these quantities, we can now turn our focus to the actual definition of a loss function that approximates the cost function defined in <ref>.

Let us first define the loss attached to a possible decision made at time $t$ for a trade-off parameter $\alpha$ value and a time series $\xuptot$ from class $y$ as

\begin{equation}
\mathcal{L}_t(\xuptot, y ; \alpha) = \alpha \mathcal{L}_c (\xuptot, y) + \left(1-\alpha\right) \mathcal{L}_e(t) \, .\label{eq:lt}
\end{equation}

The commonly used choice for $\mathcal{L}_c$ would be the logistic or cross entropy loss $\mathcal{L}_c (\xuptot, y) = -{y}\log(\yhat)$.
This loss is fully compatible with our proposed method and exponentially penalizes wrong classifications, as illustrated in <ref>.
However, we would like to be able to compare our model with the best state-of-the-art early classification model of <cit.> using the identical trade-off parameters $\alpha$.

When approximating the cost function defined in <ref>, we will use

\begin{equation}
\mathcal{L}_e(t) = \frac{t}{T}
\end{equation}

\begin{equation}
\mathcal{L}_c (\xuptot, y) = 1-\yhat^+ \, ,
\label{eq:loss:classification}
\end{equation}

where $\yhat^+$ is the probability output by the model for class $y$ when fed with input $\xuptot$.
As shown in <ref>, our $\mathcal{L}_c$ formulation is closer to the $0-1$ loss on average, hence allowing to better model the target trade-off between earliness and accuracy for a given $\alpha$ value.

Note however that <ref> suggests our loss function has lower
gradients for bad solutions (low $\yhat^+$), which means it will be weaker at preventing from bad classification decisions.
To find a compromise between optimization efficiency and comparability to <cit.>, we decided to adopt a two-phase training procedure, as described in <ref>.
In our experiments, we first pre-train the model purely for accuracy with a logistic regression loss and then subsequently fine-tune it end-to-end for accuracy and earliness using this loss formulation.

Then, we can compute a per-time-series overall loss as the expectation of $\mathcal{L}_t$ over decision timings

\begin{eqnarray}
\mathcal{L}(x, y;\alpha) &=& \mathbb{E}_{t}\left[\mathcal{L}_t(\xuptot, y ;\alpha)\right]
\label{eq:l} \\
&=& \sum_{t=0}^T P(t) \cdot \left[
\alpha \mathcal{L}_c (\xuptot, y) + \left(1-\alpha\right) \mathcal{L}_e(t)
\right]
\nonumber
\label{eq:lossentropy}
\end{eqnarray}

Finally, at classification time, for each timestamp $t$, the decision about performing classification or waiting for more samples is drawn from a Bernoulli distribution of parameter $\delta_t$.

§.§ Sequence classification models

The previous section described the loss function that we use to parameterize the probabilistic stopping decision $\delta_t$.
In this part we provide details on the classification models used throughout this paper.

We want to emphasize that the stopping output $\delta_t$ can be implemented to any differentiable model that evaluates a hidden feature vector $h_t$ at each time step $t$.
We want to demonstrate this by implementing our approach on two deep neural network topologies.

§.§.§ Recurrent Neural Network

Recurrent neural networks, for which the hidden state is computed as $h_t \leftarrow \text{RNN}(x_t, h_{t-1})$, are broadly used for temporal data processing.

Propagating gradients through the time dimension, however, is known to cause gradients to either vanish <cit.> or explode <cit.>.
Gated recurrent neural networks, such as Long Short Term Memory (LSTMs) <cit.> or Gated Recurrent Units (GRUs) <cit.> alleviate this by introducing additional gates that control the gradient flow through time.
These networks process variable length sequences by iteratively applying their parametrization on newly observed data.
Hence, extracted features contain expressive high-level information of the input data.
However, these networks are often difficult to optimize on smaller datasets.

For this work, we employ a straightforward classification strategy using stacked LSTM layers, denoted by a bracketed, raised index.
Each recurrent layer $l$ produces a hidden state $h^l_t \leftarrow \text{LSTM}(h^{l-1}_t,h^{l}_{t-1})$ based on the current output of the previous layer $h^{l-1}_t$ and time $h^{l}_{t-1}$.
The input observation is used for the first layer while the last hidden state is employed to produce classification probabilities.
We implement our approach by considering a second linear output from the last layer's hidden state and fine-tune the model end-to-end, as described in the previous section.
We will present qualitative results using this network topology in <ref>.

§.§.§ 1D Convolutional shapelet Model

Illustration of the employed Conv1d model.
A input time series is convolved with kernels (red) of increasing sizes. The resulting feature maps (blue) are then max-pooled up to time $t$.
This model can process a sequence up to time $t$, as indicated by the vertical line in the respective input, feature maps and output values.
These sequentially max-pooled values are then concatenated to one long feature vector $h_t$ from which output probabilities $\yhat_t$ and the stopping probability $\delta_t$ is evaluated.
Note that at time $t$ a single $\delta_t$ value and $\yhat_t$ vector are evaluated from $h_t$.

For a quantitative evaluation, we chose a single-layer convolutional model that is similar in spirit to shapelet models <cit.>, as illustrated in <ref>.
An input time series $x$ is shown at the top that is known up to time $t$ indicated by the vertical line.
Mono-dimensional convolution blocks $f^l = x \circledast k$ produce feature maps of varying temporal context using increasingly large kernels $k \in \mathbb{R}^{w^l \times d^l}$.

The number $l$ of 1d convolution blocks, as well as the number $d^l$ of kernels per kernel length $w^l$ are model-specific hyper-parameters.
We pad the input time sequence with zeros for the respective kernel size $w^l$ to ensure that the observation time $t$ aligns with the right end of the convolution kernel.
This padding guarantees that in our model no input data after $t$ is convolved.
To gather features for a classification decision up to a specific time $t$, sub-sequence max-pooling is used $max(f^l_{\rightarrow t})$.

The resulting max-pooled features of all convolutions are then concatenated to one combined feature vector $h_t$.
Additionally, we employ dropout and batch normalization to this feature vector to improve convergence and counteract over-fitting.
Note that sub-sequence level max-pooling allows the model to produce a hidden representation $h_t$ whatever the length of the input time series.
Based on $h_t$, probabilities for each class are evaluated by a fully connected layer with softmax activation.
Similarly, a second fully connected layer (with sigmoid activation) is employed for the probabilistic stopping rule $\delta_t$.

§ RESULTS

Now that we have defined our model, we assess its validity through extensive experiments on a diverse set of 46 publicly available time series classification datasets.
These 46 datasets are used in <cit.> who, fortunately, also report classification accuracy and earliness of compared approaches.
They are made available on the UCR repository <cit.> and cover a wide range of data modalities, ranging from motion capture through audio to Electrocardiography (ECG) sensor data.
The competitors considered here are the best method from <cit.> (denoted SR2-CF2), the pioneering work of <cit.> (ECTS), as well as EDSC <cit.>, and RelClass <cit.> methods.

[overall loss ]
[classification loss]
[earliness loss]
Accuracy and test-losses during training of one recurrent neural network model during the early classification fine-tuning phase. The trade-off factor $\alpha=0.8$ tends to equally balance earliness and accuracy.

[Input sequence $x$]
[$P(t)$ at epoch 3]

[$P(t)$ at epoch 55]
A input sample and two $P(t)$ scores of one particular sample. This model was trained using the trade-off factor $\alpha=0.8$. At training on accuracy and earliness, we initialize $\delta_t$ so that $P(t)$ focuses on late observations, as shown in <ref>. After the training period $P(t)$ has focused on $t=85$ which corresponds to a characteristic feature in the input sequence.

§.§ Training Procedure

As presented in <ref>, our approach is designed to be easily plugged on a pre-trained shapelet-like model.
To simulate this in our experiments, we train our network in two distinct phases.
The classification phase is the pre-training based on a standard cross-entropy loss function.

After this pre-training step, we branch our earliness module parametrized by $\theta_\delta$ and fine-tune all model parameters using the loss function defined in <ref>.
These two learning phases differ in three main points.
First, a term that penalizes for late classifications is introduced in the second phase.
Second, the way we train our model using time series of varying lengths during phase 1 is equivalent to considering a uniform law for $P(t)$, while, during phase 2, this is adapted to lower the delay penalization term in the loss.
Third, in phase 2, the classification-based part of the loss is no longer a cross-entropy loss but rather a linear one so as to better match the trade-off between earliness and accuracy as defined by the target cost function.

§.§ Model selection

As stated above, the 46 datasets that are used in these experiments are diverse in nature, which makes the selection of hyper-parameters a critical task.
We select the network configuration through three-fold cross validation after $30$ epochs of training on pure accuracy.
The parameter space was evaluated using grid search.
The tuned hyper-parameters for the convolutional model include the number of convolution blocks $l \in \{4,6,8\}$, the number of kernels per convolution $d^l \in \{50, 75, 100\}$, the increase rate of kernel width per layer $\delta_w \in \{30, 50, 70\}$.
As for the hyper-parameters for the recurrent model, we varied the number of recurrent layers $n \in \{2,3,4,5,6\}$ and the number of hidden features per layer $r \in \{64,128,256,512\}$.
For both models, we searched with two learning rates $\eta \in \{0.1, 0.01\}$ for our Adam optimizer <cit.>.

To counteract over-fitting we choose a dropout rate of 50We decided to perform the model-specific hyper-parameter tuning solely based on the pure classification phase, to separate model-specific hyper-parameters related to classification performance from parameters related to earliness.
Note that this separate tuning strategy conveniently follows the spirit of fine-tuning an existing model with our proposed stopping rule.

§.§ Qualitative Results

In this section we present qualitative results obtained from the recurrent model described in <ref>.

First, we want to get an impression on the training process of a model that was previously trained on pure accuracy and is now additionally optimized for earliness.
To demonstrate our approach, we selectively chose the TwoPatterns synthetic dataset from the UCR datasets.
We present quantitative results on all UCR datasets in the next section.
Here, we show the accuracy and loss functions on the test dataset during the training process in <ref>.
Following <ref>, the overall loss, shown in <ref> is a $\alpha=0.8$ weighted sum of classification loss in <ref> and earliness loss <ref>.
By observing the $P(t)$ scores of a single input sample at selected epochs in <ref> we gain a complete picture of the visible dynamics in the training process.
As stated before, we initialized the parametrization of the stopping probability $\theta_{\delta}$ such that late classifications are favored in the first training steps.
This can be observed in <ref> where $P(t)$, and thus $\delta_t$ focus solely on the last observation.
This is confirmed by the earliness loss during training in <ref> that indicates a large loss in the initial epochs.
After few training epochs on earliness and accuracy, one can observe how the earliness loss falls to a lower, stable level.
When cross-referencing the example in <ref> at epoch 55, one can see that the $P(t)$ has reached a stable position at time $t=85$.
This time corresponds to a classification-characteristic feature in the input time series in <ref>.
The classification loss in <ref> and accuracy <ref> during the entire training process show that the classification accuracy remained on a stable level.
Overall, this qualitative example has demonstrated how a recurrent neural network can be retroactively optimized for performing earlier classifications without loss in accuracy.

§.§ Quantitative state-of-the-art comparison

Earliness weight parameters for other early classification methods. Each row represents the parameter for one earliness step. The accuracies and earliness values for each UCR dataset run have been publicly provided by <cit.>.

Ours     SR2-CF2     RelClass     EDSC     ECTS

$\alpha=0.6$     $\alpha=0.6$     $t=0.001$     $t=2.5$     $\text{s}=0.1$

$\alpha=0.7$     $\alpha=0.7$     $t=0.1$     $t=3.0$     $\text{s}=0.2$

$\alpha=0.8$     $\alpha=0.8$     $t=0.5$     $t=3.5$     $\text{s}=0.4$

$\alpha=0.9$     $\alpha=0.9$     $t=0.9$     $t=3.5$     $\text{s}=0.8$

Qualitative examples on varying earliness factors from our convolutional model in blue compared to the SR2-CF2-variant of <cit.> in orange.

Average decision time as $\frac{t_\text{stop}}{T}$ ]

Overall cost]

Average decision time as $\frac{t_\text{stop}}{T}$ ]

Overall cost]

Scatter plots of our method to its closest competitor SR2-CF2 on all 46 UCR datasets. Different earliness factors are illustrated by colored points. The blue shaded area indicates a better performance of our model, while the red shaded area indicates a worse performance.

In this section we quantitatively compare our early classification method with the other early classification approaches presented in <ref>.
As stated above, <cit.> provide accuracy and earliness results for their and comparative methods in their supplementary material
We can hence compare our results on a diverse collection of datasets with a large number of early classification approaches.
However, we would like to emphasize that a comparison of these early classification methods requires care in terms of evaluation criteria and choice of earliness parameters.
The trade-off between earliness and accuracy is parametrized differently in each approach.
We followed the parameters evaluated by <cit.>, as shown in <ref>.
In contrast to the other methods, we carefully designed our loss function and trade-off parameter $\alpha$ to be numerically comparable with <cit.>.
To enable this, we avoided the commonly used cross entropy loss and employed a more comparable linear loss, as shown in <ref> and illustrated in <ref>.

Also, we compare methods based on two criteria, earliness and accuracy, that should be jointly evaluated.
Considering both objectives separately is not sufficient for a fair evaluation, since the overall goal is to minimize the mixed cost function defined in <ref>.
Evaluating on this cost function when comparing to SR2-CF2 is a fair and natural way to compare performance since this cost is the explicit quantity that both our method and its baseline aim at minimizing.
Hence we calculate an $\alpha$-weighted score similar analogous to the loss function in <ref>.
This score is closest to the objective defined to the respective approaches and should be the primary evaluation focus.

Since we evaluate this score on a large number of datasets, we decided to use win/loose plots for a first quantitative evaluation.
Datasets where our method achieved a higher scores than the competitor are counted to the win category and vice-versa.
We illustrate the results in <ref>.

Domination table of state-of-the-art approaches in early classification compared to this work.

$\alpha$     SR2-CF2     RelClass     EDSC     ECTS

$0.6$     7 / 38     31 / 14     34 / 8     40 / 5

$0.7$     3 / 42     28 / 17     28 / 14     35 / 10

$0.8$     5 / 40     23 / 22     30 / 12     34 / 11

$0.9$     12 / 33     19 / 26     33 / 9     26 / 19

Our approach clearly outperforms the early classification models EDSC, RelClass, and ECTS on almost all evaluated trade-off factors.
However, it seems that <cit.> achieve a better score on many datasets with their non differentiable early stopping heuristic.

We further analyze the differences between our approach, SR2-CF2 of <cit.> and the RelClass <cit.> method in <ref>.
Here we show accuracy, earliness and $\alpha$ weighted scores as scatter plots.
In this figure, the horizontal axis shows our performance, while the vertical axis is based on scores reported by <cit.> and <cit.>, respectively.
Each dot in these plots corresponds to a single dataset for a given $\alpha$ value.
Colors indicate different target trade-offs (i.e. $\alpha$ values).
The diagonal marks equal performance between the approaches.
It seems that our approach achieves similar accuracies compared to SR2-CF2, as shown in <ref>, but performs the classifications later.
This results in an overall score that is slightly worse, as shown in <ref>.

Even though our evaluated scores are slightly worse than the numbers reported by <cit.> we present a method that can be broadly employed, as we demonstrate on two separate classification models, as described in <ref>.
In contrast to SR2-CF2, our method is scalable for large datasets, requires only a single model parametrization to process a variable length sequence, and is trained via standard back-propagation and gradient descent.
In contrast, SR2-CF2 <cit.> trained separately a Gaussian process classifier for each possible sequence length and designed a binary, non-differentiable stopping decision that requires a parametrization using genetic algorithms.

Still, our approach clearly outperformed the baseline methods ECTS and EDSC and achieves better scores compared to relclass in <ref>.
Comparing to RelClass in particular in <ref>, one can observe the variability in terms of earliness of the classification between the approaches.
Overall, our method is comparable and competes well with the state-of-the-art in early classification while being easier deployable and more applicable to the broader machine learning community by relying on a pure end-to-end optimization with standard gradient back-propagation.

§.§ Influence of the Earliness Accuracy trade-off factor

In our final experiment we qualitatively assess the effect of the accuracy-earliness trade-off parameter $\alpha$ for our approach and the SR2-CF2 <cit.> method.
As stated previously, we specifically designed our loss function, see <ref>, such that the effect of $alpha$ on earliness and accuracy is similar.

In <ref>, we show both approaches on the two datasets ECGFiveDays and Haptics. The SR2-CF2 method is illustrated in orange, while our approach is shown in blue.
Accuracy and earliness for different values of $\alpha \in \{0.6, 0.7, 0.8, 0.9\}$ are drawn on the axis.
As intended, an increasing weight on the classification loss with $\alpha \in \{0.8,0.9\}$, improves the achieved accuracy, while simultaneously increasing the earliness, as can be observed consistently for both approaches.
Comparing our approaches, it seems that our method tends towards late classifications at the end of the sequence when a large weight on accuracy $\alpha=0.9$ is chosen.
We suspect that the initialization of $\delta_t$ for late classifications in the initial training epochs can introduce a bias towards later the overall classification times which could explain this behavior.
Overall, even though we taylored our method towards comparability with <cit.>, there seems to be a slighly different effect of $\alpha$ on overall accuracy and earliness.
This demonstrates the difficulties involved in comparing different approaches on these two antagonistic objectives.
Summarizing, these results show that the trade-off parametrization of earliness and accuracy is an important parameter that will have be to chosen based on the user's preference.

§ DISCUSSION

In this work, we propose a novel, end-to-end trainable mechanism that produces a probabilistic decision that can be used to prematurely stop evaluating a sequence of observations if enough classification-relevant characteristics have been observed.
It is broadly applicable to any sequential model that evaluates a hidden representation at each time $t$.
We demonstrated the applicability by implementing our mechanism on two different sequential classification models in <ref>.
We were able to compare the accuracy and earliness of our method to other early classification approaches on a diverse set of 46 datasets that enclose various types of use-cases for time series classification.
These results have shown that our method was better or comparable to methods in the state-of-the-art in early classification.
Compared to the best early classification model of <cit.> we achieved similar accuracies, but with later classifications, thus resulting in a worse overall score.
Nevertheless, we argue that our approach is methodically simpler, more scalable to large datasets and easier to deploy in existing sequential classification methodologies.
Further we do use a single model parametrization for a variable length sequence and train our stopping rule end-to-end.
In contrast, <cit.> used a different classification model parametrization at every time step, and were forced to employ genetic algorithms to train their models due to a non-differentiable cost function.
Also, we would like to emphasize that we modified our training procedure and loss function to be able to compare to <cit.> on an equal accuracy-earliness trade-off parameter $\alpha$.
Potential adopters of our method can employ a more competitive loss function for classification, such as logistic regression, if they do not intend to strictly compare with the approach of <cit.>.


#  When does a double-layer potential equal to a single-layer one?

Alexander G. Ramm
Department of Mathematics, Kansas State University,
Manhattan, KS 66506, USA
<EMAIL_ADDRESS>
http://www.math.ksu.edu/ $\sim$ ramm

###### Abstract

MSC: 31A10; 35C15; 35J05 Key words: potetial theory

Let $D$ be a bounded domain in ${\mathbb{R}}^{3}$ with a closed, smooth,
connected boundary $S$, $N$ be the outer unit normal to $S$, $k>0$ be a
constant, $u_{N^{\pm}}$ are the limiting values of the normal derivative of
$u$ on $S$ from $D$, respectively $D^{\prime}:={\mathbb{R}}^{3}\setminus D$;
$g(x,y)=\frac{e^{ik|x-y|}}{4\pi|x-y|}$,
$w:=w(x,\mu):=\int_{S}g_{N}(x,s)\mu(s)ds$ be the double-layer potential,
$u:=u(x,\sigma):=\int_{S}g(x,s)\sigma(s)ds$ be the single-layer potential.

In this paper it is proved that for every $w$ there is a unique $u$, such that
$w=u$ in $D$ and vice versa.

Necessary and sufficient conditions are given for the existence of $u$ and the
relation $w=u$ in $D^{\prime}$, given $w$ in $D^{\prime}$, and for the
existence of $w$ and the relation $w=u$ in $D^{\prime}$, given $u$ in
$D^{\prime}$.

##  1 Introduction

Let $D$ be a bounded domain in ${\mathbb{R}}^{3}$ with a closed, smooth,
connected boundary $S$, $N=N_{s}$ be the outer unit normal to $S$ at the point
$s\in S$, $k>0$ be a constant, $g(x,y)=\frac{e^{ik|x-y|}}{4\pi|x-y|}$,
$w:=w(x,\mu):=\int_{S}g_{N}(x,s)\mu(s)ds$ be the double-layer potential,
$w^{\pm}$ are the limiting values of $w$ on $S$ from $D$, respectively, from
$D^{\prime}$, $u:=u(x,\sigma):=\int_{S}g(x,s)\sigma(s)ds$ be the single-layer
potential, $u_{N^{\pm}}$ are the limiting values of the normal derivative of
$u$ on $S$ from $D$, respectively $D^{\prime}:={\mathbb{R}}^{3}\setminus D$,
the overbar, e.g. $\bar{\sigma}$, denotes the complex conjugate of $\sigma$,
$H^{0}:=L^{2}(S)$, $H^{1}:=W^{1}_{2}(S)$ is the Sobolev space, $\mu\in H^{0}$.
We write iff for if and only if and use the known formulas for the limiting
values of the potentials on $S$, see [3], pp. 148, 153:

$w^{\pm}=0.5(A^{\prime}\mp I)\mu;\quad u_{N}{{}^{\pm}}=0.5(A\pm I)\sigma;\quad
w_{N}^{+}=w_{N}^{-};\quad Q(\sigma):=\int_{S}g(t,s)\sigma(s)ds,$ (1)

where $A^{\prime}\mu=\int_{S}g_{N_{t}}(t,s)\mu(s)ds$,
$A\sigma=\int_{S}g_{N_{s}}(t,s)\sigma(s)ds$. In this paper it is proved that
for every $w$ in $D$ there is a unique $u$, such that $w=u$ in $D$, and vice
versa.

Necessary and sufficient conditions are given for $w=u$ in $D^{\prime}$ and
for $u=w$ in $D^{\prime}$.

Let us state these results:

Theorem 1. For every $w$, defined in $D$, there is a unique $u$ such that
$w=u$ in $D$, and vice versa.

For every $w$, defined in $D^{\prime}$, there is a unique $u$ such that $w=u$
in $D^{\prime}$ iff

$\int_{S}w^{-}pds=0\quad\forall p\in N(Q).$ (2)

For every $u$, defined in $D^{\prime}$, there is a unique $w$ such that $w=u$
in $D^{\prime}$ iff

$\int_{S}urds=0\quad\forall r\in N(A+I).$ (3)

In Section 2 proofs are given. In Section 3 it is proved that $Q:H^{0}\to
H^{1}$ is a Fredholm operator.

##  2 Proofs

a) Assume that $w=w(x,\mu)$ is given in $D$. Let us prove that $u=u(x,\sigma)$
exists such that $u=w$ in $D$, and $u$ is uniquely defined by $w$. First, let
us prove the last claim. Suppose $u_{1}=w$ and $u_{2}=w$ in $D$. Let
$u_{1}-u_{2}:=u$, $u=\int_{S}g(x,s)\sigma ds$ in $D$,
$\sigma=\sigma_{1}-\sigma_{2}$. Then $u|_{S}=0$, $(\nabla^{2}+k^{2})u=0$ in
$D^{\prime}$ and $u$ satisfies the radiation condition, so $u=0$ in
$D^{\prime}$. Thus, $u=0$ in $D\cup D^{\prime}$, $u_{1}=u_{2}$, and the claim
is proved.

Let us now prove the existence of $u$ such that $w=u$ in $D$. One has

$w^{+}=0.5(A^{\prime}-I)\mu=u|_{S}=Q(\sigma)$ (4)

This is an equation for $\sigma$, $w^{+}$ is given. This equation is
equivalent to

$w^{+}=(I+Q_{1}Q_{0}^{-1})Q_{0}(\sigma)=(I+Q_{1}Q_{0}^{-1})\eta,\quad\eta:=Q_{0}(\sigma),\quad\sigma=Q_{0}^{-1}\eta.$
(5)

The operator $Q_{1}Q_{0}^{-1}$ is compact in $H^{0}$, see Section 3. Therefore
a necessary and sufficient condition for the solvability of equation (5), and
the equivalent equation (4), is:

$\int_{S}w^{+}\bar{\eta}ds=0\quad\forall\eta\in
N((I+Q_{1}Q_{0}^{-1})^{\star}),$ (6)

where $N(B)$ is the null space of the operator $B$ and $B^{\star}$ is the
adjoint operator to $B$ in $H^{0}$, $B=I+Q_{1}Q_{0}^{-1}$, $B$ is of Freholm
type in $H^{0}$. The kernel $g(t,s)$ of $Q$, the function
$g(t,s)=\frac{e^{ik|t-s|}}{4\pi|t-s|}$, is symmetric: $g(t,s)=g(s,t)$.
Therefore, the kernel of $Q^{\star}$ is $\bar{g}(t,s)$. Clearly,
$(I+Q_{1}Q_{0}^{-1})^{\star}\eta=0$ iff
$(I+\bar{Q}_{1}\bar{Q}_{0}^{-1})\eta=0$, or, taking the complex conjugate,
$(I+Q_{0}^{-1}Q_{1})\bar{\eta}=0$. Applying the operator $Q_{0}$ to this
equation one gets an equivalent equation $(Q_{0}+Q_{1})\bar{\eta}=0$ since
$Q_{0}$ is an isomorphism. Let $u=u(x,\bar{\eta})$. Then $u|_{S}=0$. Since
$(\nabla^{2}+k^{2})u=0$ in $D^{\prime}$ and $u$ satisfies the radiation
condition, it follows that $u=0$ in $D^{\prime}$ and
$\bar{\eta}=u^{+}_{N}-u^{-}_{N}=u^{+}_{N}$. Therefore, using the Green’s
formula, one obtains:

$\int_{S}w^{+}\bar{\eta}ds=\int_{S}w^{+}_{N}uds=0,$

and, since $u=0$ on $S$, it follows that

$\int_{S}w^{+}\bar{\eta}ds=0\quad\forall\eta\in
N((I+Q_{1}Q_{0}^{-1})^{\star}).$ (7)

Thus, the necessary and sufficient condition (6) for the solvability of (4) is
always satisfied. Therefore, $u(x,\eta)=w(x,\mu)$ in $D$,
$Q_{0}^{-1}\eta=\sigma$.

b) Asume now that $u$ is given in $D$ and prove the existence of a unique $w$
such that $u=w$ in $D$. Let us prove that $w$ is uniquely determined by $u$ if
$u=w$ in $D$. Indeed, assume that there are two $w_{j}$, $j=1,2,$ such that
$u=w_{j}$ in $D$. Then $w:=w_{1}-w_{2}=0$ in $D$. Therefore $w_{N^{+}}=0$ on
$S$. It is known (see [3], p. 154) that $w_{N^{+}}=w_{N^{-}}$, so
$w_{N^{-}}=0$. Therefore, $(\nabla^{2}+k^{2})w=0$ in $D^{\prime}$,
$w_{N^{-}}=0$ on $S$, and $w$ satisfies the radiation condition at infinity.
This implies $w=0$ in $D^{\prime}$, so $w=0$ in $D\cup D^{\prime}$. Therefore
$\mu=w^{-}-w^{+}=0$, so $w_{1}=w_{2}$ if $u=w_{j}$ in $D$. We have proved that
$w$ is uniquely determined by $u$ if $u=w$ in $D$.

Now the existence of the solution $\mu$ to equation (4) and the relation $u=w$
in $D$ should be proved. The operator $A^{\prime}-I$ is Fredholm in $H^{0}$,
so a necessary and sufficient condition for the equation (4) to be solvable
is:

$\int_{S}Q(\sigma)\bar{h}ds=0\quad\forall h\in N((A^{\prime}-I)^{\star}).$ (8)

One has $(A^{\prime}-I)^{\star}h=(\bar{A}-I)h=0$ iff $(A-I)\bar{h}=0$. If
$(A-I)\bar{h}=0$, then $u_{N}^{-}(s,\bar{h})=0$, so $u(s,\bar{h})=0$ in
$D^{\prime}$. Note that $u(s,\bar{h})=Q(\bar{h})$. Therefore, $Q(\bar{h})=0$
in $D^{\prime}$. Since $Q$ is a symmetric operator in $H^{0}$, one has:
$\int_{S}Q(\sigma)\bar{h}ds=\int_{S}\sigma Q(\bar{h})ds=0$. Consequently,
condition (8) is always satisfied, the solution $\mu$ to equation (4) exists
and $u=w$ in $D$.

c) Assume that $w$ is given in $D^{\prime}$. Let us prove that $u$ exists such
that $u=w$ in $D^{\prime}$ iff

$\int_{S}w^{-}\bar{p}ds=0\quad\forall p\in N(Q^{\star}).$ (9)

Consider the equation for $\sigma$:

$w^{-}=Q(\sigma).$ (10)

The operator $Q:H^{0}\to H^{1}$ is Fredholm-type. Thus, a necessary and
sufficient condition for the solvability of the above equation is equation
(9). If $\sigma$ solves (10), then $u(x,\sigma)=w$ in $D^{\prime}$ because the
value of $u$ on $S$ determines uniquely $u$ in $D^{\prime}$.

d) Assume now that $u$ is given in $D^{\prime}$. Let us prove that $w$ exists
such that $u=w$ in $D^{\prime}$ iff

$\int_{S}u\bar{p}ds=0\quad\forall p\in N(\bar{A}+I).$ (11)

Note that $p\in N(A+I)$ iff $\bar{p}\in N(\bar{A}+I)$. The equation for $\mu$,
given $u$ in $D^{\prime}$, is:

$0.5(A^{\prime}+I)\mu=u.$ (12)

Since the operator $A^{\prime}+I$ is Fredholm in $H^{0}$, a necessary and
sufficient condition for the solvability of (12) for $\mu$ is:

$\int_{S}u\bar{p}ds=0\quad\forall p\in N((A^{\prime}+I)^{\star}).$ (13)

If $p\in N((A^{\prime}+I)^{\star})$, then $p\in N(\bar{A}+I)$, so condition
(11) is the same as (13). As in section c), the relation $u=w$ in $D^{\prime}$
is a consequence of the fact that $u=w^{-}$ on $S$.

Theorem 1 is proved. $\Box$

##  3 Auxiliary lemma

Recall that $H^{1}$ is the Sobolev space on $S$, $H^{0}=L^{2}(S)$.

Lemma 1. The operator $Q=Q_{0}+Q_{1}:H^{0}\to H^{1}$ is of Fredholm-type,
where $Q_{0}$ is the operator with the kernel $\frac{1}{4\pi|t-s|}$ and
$Q_{1}$ has the kernel $\frac{e^{ik|t-s|}-1}{4\pi|s-t|}$. The operator $Q_{0}$
is an isomorhism of $H^{0}$ onto $H^{1}$, which has a continuous inverse. The
operator $Q_{1}Q_{0}^{-1}$ is compact in $H^{0}$.

Proof. Let us check that $Q_{0}:H^{0}\to H^{1}$ is an isomorphism. The Fourier
transform of the kernel $\frac{1}{4\pi|x-y|}$ is positive:
$\int_{{\mathbb{R}}^{3}}\frac{e^{i\xi\cdot
x}}{4\pi|x|}dx=\frac{1}{|\xi|^{2}}$. So, $Q_{0}:H^{0}\to H^{1}$ is injective.
The kernel of the operator $Q_{1}$ is smooth enough for $Q_{1}:H^{0}\to H^{1}$
to be compact. Let us check that $Q_{0}:H^{0}\to H^{1}$ is surjective. Let
$f\in H^{1}$ and $Q_{0}\sigma=f$. Then $u:=u(x,\sigma)=\int_{S}g(x,s)\sigma
ds$ solves the problem: $\nabla^{2}u=0$ in $D$, $u|_{S}=f$. By the known
elliptic estimates (see, e.g., [1]) one has
$\|u\|_{H^{3/2}(D)}\leq\|u\|_{H^{1}(S)}$. Therefore, $\nabla u\in H^{1/2}(D)$
and, by the trace theorem, $u|_{S}\in H^{0}(S)$. This proves surjectivity of
$Q_{0}:H^{0}\to H^{1}$. Thus, $Q_{0}$ is an isomorphism of $H^{0}$ onto
$H^{1}$ which has a continuous inverse. The sum of an isomorphism $Q_{0}$ and
a compact operator $Q_{1}$ is a Fredholm operator, see, e.g., [2]. The
operator $Q_{1}Q_{0}^{-1}$ is compact in $H^{0}$ because the kernel of $Q_{1}$
is sufficiently smooth. Although the operator $Q_{1}Q_{0}^{-1}$ is defined on
a dense subset $H^{1}$ of $H^{0}$, but since this operator is bounded in
$H^{0}$ its closure is a bounded operator in $H^{0}$. Since the kernel of
$Q_{1}$ is $O(|s-t|)$, the kernel of $Q_{1}Q_{0}^{-1}$ is a continuous
function of $|s-t|$ and the surface $S$ is a compact set. Therefore, the
operator $Q_{1}Q_{0}^{-1}$ is compact in $H^{0}$.

Lemma 1 is proved. $\Box$

Competing Interests: the author does not have such interests; this work was
not supported financially by any source.

## References

  * [1] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, Berlin, 1983.
  * [2] A.Kirillov, A. Gvishiani, Theorems and problems in functional analysis, Springer Verlag, Berlin, 1982.
  * [3] A.G.Ramm, Scattering of Acoustic and Electromagnetic Waves by Small Bodies of Arbitrary Shapes. Applications to Creating New Engineered Materials, Momentum Press, New York, 2013.