11institutetext: Observatório Nacional, street General José Cristino 77,
20921-400, Rio de Janeiro, Brazil
11email<EMAIL_ADDRESS>
# Wavelet theory applied to the study of spectra of trans-Neptunian objects

Souza-Feliciano  A.C 11 Alvarez-Candal  A 11 Jiménez-Teja  Y 11

###### Abstract

Context. Reflection spectroscopy in the near-infrared (NIR) is used to
investigate the surface composition of trans-Neptunian objects (TNOs). In
general, these spectra are difficult to interpret due to the low apparent
brightness of the TNOs, causing low signal-to-noise ratio even in spectra
obtained with the largest telescopes available on Earth, making it necessary
to use filtering techniques to analyze and interpret them.

Aims. The purpose of this paper is to present a methodology to analyze the
spectra of TNOs. Specifically, our aim was to filter these spectra in the best
possible way: maximizing noise removal, while minimizing the loss of signal.

Methods. We used wavelets to filter the spectra. Wavelets are a mathematical
tool that decompose the signal into its constituent parts, allowing us to
analyze the data in different areas of frequencies with the resolution of each
component tied to its scale. To check the reliability of our method, we
compared the filtered spectra with the spectra of water and methanol ices to
identify some common structures between them.

Results. Of the 50 TNOs in our sample, we identify traces of water ices and
methanol in the spectra of several of them, some with previous reports, while
for other objects there were no previous reports.

Conclusions. We conclude that the wavelet technique is successful in filtering
spectra of TNOs.

###### Key Words.:

Kuiper belt: general – Techniques: spectroscopic – Methods: data analysis

##  1 Introduction

Trans-Neptunian objects (TNOs) are icy relics left over from the planetary
accretion disk that orbit the Sun beyond Neptune. The investigation of their
properties, as remnants of the external planetesimal swarms, is essential to
understanding their formation and evolution. Investigating their surfaces by
spectroscopy is a tool for constraining the chemical properties and the
evolution processes that occurred in the solar system (Barucci et al.,, 2008).

To better understand the objects in this region we rely on observations,
mostly ground-based, of their surface properties. In particular, we use
reflection spectroscopy in the near-infrared (NIR). In general, the NIR
spectra of TNOs are difficult to interpret because of the objects’ low
apparent brightness, which causes a low signal-to-noise ratio (S/N) even in
data obtained with the largest telescopes available on the Earth. Therefore,
the spectra of these objects usually have a considerable amount of noise
making it necessary to use filtering techniques to analyze and interpret them.

In signal processing, noise is defined as that undesirable, random signal that
is mainly introduced by the processes of capture, storage, and transmission of
the data. Although inherent to the data and thus difficult to be disentangled
from them, the noise must be estimated and removed in the most efficient way
without altering the real information contained in the data. Hereafter we use
the term signal to indicate that part of the data containing real information,
and noise to describe that artificial, random signal introduced by the
mechanisms previously described.

Filters are usually applied to remove noise from signals. Due to its random
nature, noise is identified with unstructured high frequencies. Filtering
techniques try to identify these frequencies and eliminate or attenuate them.

Among the most commonly used techniques are rebinning, the runing box, and the
Fourier transform. However, these techniques are not completely efficient
because they either decrease the spectral resolution (rebinning and runing
box) or they can incorrectly remove high frequency spectral features easily
confused with noise (Fourier transform). Two important problems affect this
last technique: 1) the definition of “high frequency” is somewhat arbitrary,
and lacks a robust definition, and 2) data are processed as a whole. In this
sense, the Fourier transform provides the frequencies that globally compose
the data. There is no information on the localization of these frequencies,
that is, whether they appear throughout the whole signal or just in a certain
interval. As they are all treated equally with this global analysis, it is
difficult to determine whether a high frequency is truly related to noise or
instead a real feature of the signal, thus containing information. This second
problem is usually overcome by applying the Fourier transform in windows, but
the arbitrary definition of the size of the window and the border effects
induced by them made us choose a different technique to filter our TNO data.

Wavelets are similar to the Fourier transform since they decompose the data
into the frequencies that compose them, but with the difference that these are
localized (although not perfectly). The wavelets perform a multiscale analysis
of the data, where the different frequencies that compose the signal are
roughly located with an uncertainty that depends on the resolution of the
scale. The information provided by the wavelets is thus essentially the same
as that from Fourier, but detailed and split between different scales. With
this new approach, a high frequency is not immediately associated with noise,
but depending on its modulus (i.e., its presence and weight in the data), it
will be considered a significant piece of information or noise.

Wavelets have been used in astrophysics for various purposes: the denoise
cosmological simulations (Romeo et al.,, 2003, 2004), to remove the background
light in galaxy clusters and detect faint small objects to complete their
luminosity function (Livermore et al.,, 2017), to identify the different
scales in wavelet decomposition with luminous components in groups of galaxies
(Da Rocha et al.,, 2008; Da Rocha and Mendes de Oliveira,, 2005), and to study
mutual events among Galilean satellites (Ramírez et al.,, 2006), among others.

Here we show how the application of the wavelets to filter the data allows us
to reduce the amount of noise in the spectra of 50 TNOs obtained with the
instrument SINFONI at the Very Large Telescope in Cerro Paranal, Chile. We
describe the sample in the Sect. 2 and the chosen fitering technique in Sect.
3. The methodology used is shown in Sect. 4 and the results are presented in
Sect. 5. Finally, the conclusions are drawn in the Sect. 6.

##  2 Data

All spectra shown in this paper were obtained with the Spectrograph for
Integral Field Observations in the Near-Infrared
(SINFONI)111http://www.eso.org/sci/facilities/paranal/instruments/sinfoni.html
located in the Cassegrain focus of the unit 4 of the Very Large Telescope
(VLT). SINFONI provides a spectroscopic imaging in 3-D in the spectral range
between 1.05 and 2.45 $\mu$m. This instrument has four different modes of
observation, J, H, K, and H+K, with a spectral resolution of 1500 in the H+K
band. The spectra of the sample were obtained using the H+K mode because it
allows us to observe the H and K bands simultaneously. The spatial resolution
was 0.25 arcsec/spaxels, with corresponding field of view of 8.0 arcsec.

Our sample was selected from the objects observed within an ESO Large Program
carried out between 2007 and 2009 (P.I.: M. A. Barucci). The objects analyzed
are listed in Table 1. Most of the spectra presented here have already been
published. Details of those marked with (+) can be found in Barucci et al.,
(2011), with (++) in DeMeo et al., (2010), with (–) in Barucci et al., (2010),
with (#) in Barucci, M. A. et al., (2012), with (*) in Guilbert et al.,
(2009), with (-) in Alvarez-Candal et al., (2007), (**) in Merlin et al.,
(2007), and Pluto and Charon in Merlin et al., (2010). We note that the
spectra of Chiron and 1996 TO66 have not been previously published;
nevertheless, the details of their reduction can be found in any of the
previous references. In this way, according to the classification of Gladman
et al., (2008), among the spectra in our sample are included 13 classical
objects, 10 resonant objects, 16 centaurs, 4 scattering disk objects, and 5
detached objects.

The sizes of the objects range from 20 to 2700 km (see Table 1), several of
which with diameters over 500 km (Pluto, Triton, Haumea, Quaoar, etc.). From a
technical point of view, it is easier to detect absorption features in the
spectra of the largest objects, due to their S/N and also because large
objects can retain a larger reservoir of volatiles (Schaller and Brown,,
2007). Smaller objects, which constitute most of our sample, are more
difficult to deal with because (1) they are too small to have kept the
volatiles that were accreted onto them when they were formed (Schaller and
Brown,, 2007); and/or (2) of their small sizes, their apparent brightness is
too faint and therefore the S/N is too low to detect small absorption
features.

Unfortunately, nothing can be done in case (1), but we can tackle case (2) and
try to work with the spectra generated by these objects through the
application of techniques that improve their signal-to-noise ratios until the
arrival of the next generation of telescopes. Therefore, we applied the
wavelets to filter all objects in our sample. For large TNOs (d $>$ 500 km)
(Brown,, 2012), especially those that have been well studied in the
literature, the goal was to verify whether the application of wavelets can be
a tool for reducing the noise and for recognizing the absorption bands already
reported in the literature without losing information. As the result was
positive, we applied the same technique to smaller TNOs (d $<$ 500 km) and
tried to identify possible evidence of absorption bands of water and methanol
ices that the noise could be hiding.

##  3 Data processing

As mentioned above, due to the overall faintness of the TNOs, their spectra
usually have relatively low S/N (seldom over 20 or 30). It is therefore
necessary to use a filtering technique to analyze and interpret them.

The main purpose of our study is to use the wavelets technique to reduce the
maximum amount of noise without changing the shapes and positions of the
possible absorption bands of the spectra, if present, in order to identify
them. Once filtered, some spectral parameters are calculated to estimate the
possibility of absorption detection, due to some ices of astrophysical
interest such as water and methanol ices. After that, these spectra are
compared to spectra calculated from optical constants measured in the
laboratory for the ices mentioned. This step is necessary to link the
absorptions detected in the filtered spectra with the ices we are looking for.

###  3.1 Wavelets

The wavelet transform is a multiresolution technique which decomposes the data
into their constituent frequencies, roughly located in time (or wavelength)
and frequency simultaneously. It is based on the so-called wavelet function or
mother wavelet $\psi$, which can be any squared-integrable function with zero
mean:

$\int_{-\infty}^{+\infty}\psi(t)dt=0.$ (1)

These two simple properties guarantee that $\psi$ has an undulating shape, and
is the reason why it is called a wavelet. For each wavelet $\psi$, we can
build its associated wavelet family, consisting of the original wavelet
function $\psi$ dilated and translated by different factors, the scale $j$ and
the translation factor $k$, respectively:

$\psi_{j,k}=\frac{1}{\sqrt{2^{j}}}\psi\left(\frac{t-2^{j}k}{2^{j}}\right).$
(2)

It can be proved that these wavelet families are mathematically orthonormal
bases of the $\mathcal{\\{}L\\}^{2}$-space of the squared integral functions.
This means that any function (or data), provided that it is smooth enough, can
be decomposed as the linear combination of the elements of the wavelet family

$f(t)=\sum_{j=0,j\in\mathbb{Z}}^{\infty}\sum_{k\in\mathbb{Z}}d_{j,k}\psi_{j,k}(t),$
(3)

where the coefficients $d_{j,k}$ are the wavelet coefficients. These
coefficients are calculated according to the following expression:

$d_{j,k}=\int_{-\infty}^{+\infty}f(t)\overline{\psi_{j,k}(t)}dt.$ (4)

The wavelet coefficients are related to the details (frequencies) that compose
the data. Their lower scales ($j$) give information on the finer details that
compose the data, such as high frequency structures, pseudo-impulsive
features, or noise. These fine details are very well localized in time due to
the small support size of the associated wavelet functions $\psi_{j,k}$ (its
width in the time space). Higher $j$ scales tell us about coarser details, low
frequency structure in the data, usually related with information. Due to the
increase in the support size of the $\psi_{j,k}$, these coarse details are
localized in time with a higher uncertainty. However, the support size of a
function in time is inversely proportional to its support size in the Fourier
domain. That means that the narrower the $\psi_{j,k}$ are, the wider their
support in frequency becomes, thus increasing the uncertainty in frequency.
For this reason, in the wavelet decomposition the high frequencies that
compose the data are not very well identified, although exceptionally well
localized in the data (i.e., we know where they are, but not who they are),
whereas the low frequency structures are better defined, but we do not know
exactly where they appear (i.e., we know who they are, but not where they
are). So, if Fourier transform provides the exact decomposition in frequency
of the data independently of its localization, the wavelet transform tells us
approximately which frequencies compose the data and approximately the
intervals where they appear. We lose precision in frequency to gain some
information on time.

This is very useful for disentangling the noise from the signal. In the
wavelet theory, the noise is no longer related to the high frequencies, but to
those structures that do not have a strong weight in the decomposition in a
certain time interval. For example, the high frequency components of a narrow
absorption line would be kept as signal (information) since these high
frequencies contain structural information of the data in that interval.
However, these same high frequencies can be identified as noise in any other
interval of the data where the subjacent structure is smoother, like a broad
line in a noisy spectrum, since the weight of these high frequencies is small
compared to that of the low frequencies. Thus, the noise will be defined as
the high frequency in a certain interval which does not have a relative
importance for the structure of the data in that interval, identified by the
modulus of its associated wavelet coefficient $d_{j,k}$. To quantify this
idea, Donoho and Johnstone, (1994) proved that when there is Gaussian noise,
there is a certain threshold (called the universal threshold) below which a
wavelet coefficient has a very high statistical probability of carrying noise.
This universal threshold for a signal of size $N$ polluted with white noise of
variance $\sigma^{2}$ is defined as

$T=\sigma\sqrt{2\log N}.$ (5)

The only problem of the universal threshold is that it assumes the variance of
the noise to be known. As this is not usually the case with real data, it can
be proved that the following parameter is a robust estimator of the variance
$\sigma^{2}$ of the noise,

$\sigma=\frac{M}{0.6745},$ (6)

where $M$ is the median of the absolute moduli of the wavelet coefficients in
the first scale.

Thus, to filter our noisy TNO spectra we decompose them up to a certain scale
$j$ using a certain wavelet family, calculate the corresponding universal
threshold $T$, and set to zero all the coefficients with $|d_{j,k}|<T$. This
is called a “hard filter” and it guarantees that the characteristics of the
signal, in our case the shape and depth of the absorption bands in our
spectra, are preserved after removing the noise.

###  3.2 Selection of the wavelet

There are several wavelet functions (each with its associated wavelet family)
that lead to different wavelet decompositions. The wavelets functions can be
real or imaginary, have peaks or be smooth, be symmetric or asymmetric, be
more or less oscillating, etc. Therefore, there is no “perfect” wavelet
function; we must choose the one that works best for our problem according to
two main criteria, the characteristics of the data and our final purpose.

As the wavelet decomposition is just the result of the convolution of the data
with the mother wavelet, dilated and translated by different factors, wavelet
functions that are similar to the data perform better. In this sense, if our
data have symmetric features (e.g., the absorption lines in our spectra) a
symmetric wavelet is preferred. A smooth, not very oscillating wavelet would
also be desirable in our case, since the spectra do not present sharp peaks or
highly oscillating features (e.g., a seismic signal).

The final goal of the wavelet decomposition is also very important for the
selection of the wavelet. Wavelets can be used to detect singularities in the
data, for instance sharp peaks, extrema, inflection points (see, e.g., Ramírez
et al., 2006). For this purpose, the support size of the wavelet is crucial
since wavelets with wide support introduce too much uncertainty in time, and
the singularity will not be tightly identified. When we are interested in
filtering the data, the support size of the wavelet is not essential (although
a narrow wavelet function is always desirable), but its compression capacity
is much more important. The wavelet transform keeps the energy of the data,
meaning that the norm of the signal is the same as the norm of the wavelet
coefficients of its decomposition. As the filtering algorithm is based on the
comparison of the modulus of the wavelet coefficients carrying noise with the
modulus of the coefficients associated with the signal, a large difference
between these two sets would be convenient. If the energy of the signal were
split into many coefficients, this comparison might not be clear and a
coefficient containing signal could be misidentified as noise. However, if the
energy of the information were concentrated in just a few wavelet
coefficients, the filtering process would be straightforward. The concept of
vanishing moments helps us know the capacity of concentration (compactness of
the decomposition) that has the wavelet. In practice, the longer the number of
vanishing moments that the function has, the fewer the coefficients needed in
the decomposition (Eq. 3).

Therefore, to select the wavelet that best filters our data, up to a certain
scale $j$, we can follow two paths: (i) perform the filtering for different
wavelets and different scales for each input spectrum one by one and decided
manually which is the best or (ii) devise an automatic algorithm. The first
case might work for one or two spectra, but it is highly inefficient for a
larger number, therefore we will follow path (ii).

We devised a blind test in order to scan among the discrete wavelets offered
by the PyWavelet package of Python with changing scales between 1 and 8. As a
test sample, we used 41 spectra of water ice with signal-to-noise ratios
ranging from about 1 to about 200, with emphasis in the range 1–50.

In order to select the optimal wavelet we use a quadratic sum of three
different figures of merit (FoMs, taken from Meza et al., 2010). All of them
are blind, i.e., with no a priori information regarding the data (e.g., what
ice lies underneath the noise.)

Before continuing we define our input spectrum (data) as $I=[I_{i},...,I_{M}]$
and the filtered spectrum (signal) as $F=[F_{i},...,F_{M}]$.

The first FoM is the roughness (adapted from Hayat et al., 1999),

${\rm\rho}=\sum_{i=1}^{M}{|[-1,1]*F_{i}|}/\sum_{i=1}^{M}{|F_{i}|},$ (7)

which measures the roughness of a signal. The asterisk stands for convolution.
A constant, smooth signal will have $\rho=0$, while increasing pixel-to-pixel
variation will have larger values of $\rho$.

A second FoM is the noise reduction (NR, adapted from Chen et al., 2003),

${\rm NR=N_{0}/N_{1}},$ (8)

where N0 is the average of the power spectrum of the data and N1 is the
average of the power spectrum of the signal. The best filtering is obtained
when NR is high.

The last FoM we use is the residual non-uniformity (RNU, adapted from Wang et
al., 2008),

${\rm RNU}={\rm
100/\overline{F}\sqrt{(1/M\times\sum_{i=1}^{M}{(F_{i}-\overline{F})^{2}})}},$
(9)

where we search for the most uniform filtered signal, and therefore for low
values of RNU. The overlined quantities are mean values.

In order to define our metric we first invert NR to search for its minimum and
renormalize all FoMs to the interval [0,1]. Then we define our blind figure of
merit (bFoM) as

${\rm bFoM}=\sqrt{\sum_{i=1}^{3}({\rm FoM_{i}})^{2}/3};$ (10)

as such, the optimal wavelet and scale for filtering will be given by the
minimum of bFoM.

By construction we know the input signal-to-noise ratio, S/No, of the spectra
in the test sample. Each one was filtered with 58 different wavelets, in eight
different scales, amounting to 454 filtered spectra per input spectrum.
Considering that we have 41 modeled spectra, in the end we have 19014 filtered
spectra each with a value of bFoM. To help visualize the results we created
maps that show a clear trend to larger gains in S/N with decreasing values of
bFoM (Fig. 1). Therefore, we adopt as our criterion for selecting the wavelet
and scale the one that minimizes bFoM; as such, we are selecting the wavelet
and scale that optimally balances the smoothness, uniformity, and noise
removal of the data. Upon visual inspection of the signal (the filtered
spectrum), the results are satisfactory.

Figure 1: Maps of the distribution of the bFoM compared to the relative gain
in S/Nw. The right panel shows data with S/No ¡ 20, while the left panel shows
data with S/No ¿ 20. (See text.)

It must be mentioned that although low values of bFoM ensure good gains in
S/N, it does not necessarily maximizes it; this is a trade off between knowing
a priori what the “noiseless” spectrum should look like and no a priori
assumptions.

Figure 1 has been split into two panels. The left panel shows the maps for
model spectra with S/No¡20, while the right panel does it for S/No¿20. We have
chosen this value as the average S/N of the spectra in our sample (see Table
1) and to illustrate that the performance of bFoM is not correlated by the
S/No of the spectrum.

We also computed the performance of a typical Fourier filtering, applying a
fast Fourier transform (S/Nf), against the filtering with the optimal wavelet
(S/Nw). It can be seen in Fig. 2 that the optimal wavelet clearly outperforms
the Fourier filtering even for low S/No spectra. It is also clear that S/Nw is
at least a factor of 2 better than the S/No.

Figure 2: Comparison between the performances of the optimal wavelet, found by
our algorithm, and a Fourier filtering. The blue squares show the ratio of S/N
between wavelet and Fourier filtering, while the red circles show the S/N
ratio between the wavelet filtering and the original data.

Therefore, in what follows, we always use the wavelet and scale that minimizes
the bFoM and perform a visual inspection to corroborate the quality of the
filtering.

##  4 Methodology

We applied the algorithm mentioned above to our 50 spectra. The signal-to-
noise ratio before and after the filtering are shown in Table 1.

In the filtered spectra we measured two absorption band depths, the first at
2.0 $\mu$m, which gives an idea about the presence of water ice, and the
second at 2.27 $\mu$m related to methanol (e.g., Merlin et al., 2012). Fluxes
were computed as the median value between 1.98–2.02, 2.18–2.22, and 2.25–2.29
$\mu$m. The error assigned to each value comes directly from the standard
deviation of the points used to compute the depth of the absorption band in
each interval. These results are presented in Table 1.

Finally, we compare the filtered spectra with laboratory ices of water and
methanol ices, obtained using the Hapke, (1981) model and optical constants
from Quirico and Schmitt, (1997). The objective was to confirm the presence of
these ices in the objects where their presence was reported, in order to
validate our technique. We also looked for evidence of these ices in the
spectra of TNOs that were previously neglected because of their low signal-to-
noise ratios.

Number | Wavelet | Scal | D${}_{2.0}\mu m$ (%) | D${}_{2.27}\mu m$ (%) | Diameter (km)111The diameters shown here were taken from the survey “TNOs are cool” (http://public-tnosarecool.lesia.obspm.fr/). | S/No | S/Nw
---|---|---|---|---|---|---|---
2060 Chiron | Bior 5.5 | 7 | 3.74 $\pm$ 0.07 | 13.3 $\pm$ 1.3 | 215.6 | 56.1 | 357.6
5145 Pholus + | Rbio 3.9 | 5 | 7.7 $\pm$ 0.4 | 33.9 $\pm$ 9.2 | 119 | 8.5 | 38.2
10199 Chariklo * | Bior 5.5 | 5 | 6.8 $\pm$ 0.2 | 1.84 $\pm$ 0.09 | 231 | 86.2 | 179.5
15874 1996 TL66 *,+ | Bior 5.5 | 8 | 22.7 $\pm$ 12.8 | 35.1 $\pm$ 16.6 | 339 | 6.3 | 46.6
19308 1996 TO66 | Bior 5.5 | 7 | 33.6 $\pm$ 40.1 | 33.0 $\pm$ 37.0 | 200 | 0.95 | 9.91
26375 1999 DE9 * | Bior 5.5 | 8 | 9.22 $\pm$ 0.05 | 5.36 $\pm$ 1.04 | 311 | 10.6 | 213.0
28978 Ixion * | Rbio 3.9 | 7 | 11.5 $\pm$ 1.5 | 27.9 $\pm$ 16.8 | 617 | 16.5 | 92.2
32532 Thereus * | Bior 5.5 | 8 | 10.0 $\pm$ 1.0 | 8.9 $\pm$ 1.9 | 62 | 15.7 | 122.2
38628 Huya - | Bior 5.5 | 6 | 7.38 $\pm$ 0.43 | 21.0 $\pm$ 2.4 | 458 | 9.0 | 31.6
42355 Typhon * | Bior 5.5 | 7 | 14.5 $\pm$ 0.2 | 4.8 $\pm$ 0.8 | 185 | 18.5 | 81.2
44594 1999 OX3 + | Rbio 3.5 | 8 | 4.15 $\pm$ 0.85 | 1.46 $\pm$ 0.43 | 135 | 15.7 | 270.0
47171 1999 TC36 * | Rbio 2.6 | 7 | 8.16 $\pm$ 0.74 | 0.74 $\pm$ 0.12 | 393 | 24.7 | 185.0
47932 2000 GN171 * | Rbio 2.8 | 6 | 20.0 $\pm$ 4.0 | 21.7 $\pm$ 7.40 | 147 | 9.08 | 34.68
50000 Quaoar * | Rbio 2.8 | 8 | 28.1 $\pm$ 0.4 | 11.8 $\pm$ 0.5 | 1036 | 23.7 | 28.9
52872 Okyrhoe ++ | Sym 15 | 8 | 2.57 $\pm$ 0.07 | 0.72 $\pm$ 0.01 | 35 | 56.7 | 180.6
54598 Bienor * | Rbio 2.4 | 8 | 12.7 $\pm$ 1.9 | 25.4 $\pm$ 12.8 | 198 | 6.4 | 96.5
55565 2002 AW197 * | Db 2 | 8 | 13.8 $\pm$ 7.9 | 21.3 $\pm$ 6.4 | 768 | 16.4 | 190.2
55576 Amycus + | Bior 5.5 | 7 | 12.6 $\pm$ 0.4 | 13.5 $\pm$ 2.0 | 104 | 15.2 | 148.8
55637 2002 UX25 + | Bior 5.5 | 7 | 12.4 $\pm$ 1.3 | 38.0 $\pm$ 8.8 | 692 | 7.0 | 11.6
55638 2002 VE95 # | Bior 5.5 | 8 | 10.28 $\pm$ 0.05 | 11.3 $\pm$ 1.1 | 250 | 21.4 | 267.4
60558 Echeclus * | Bior 5.5 | 8 | 2.5 $\pm$ 0.4 | 1.2 $\pm$ 0.5 | 65 | 10.1 | 41.8
73480 2002 PN34 ++ | Bior 5.5 | 8 | 13.9 $\pm$ 2.2 | 2.0 $\pm$ 0.6 | 112 | 12.0 | 95.7
83982 Crantor * | Db 18 | 8 | 14.0 $\pm$ 1.8 | 34.3 $\pm$ 11.0 | 59 | 15.8 | 141.9
90377 Sedna - | Bior 5.5 | 7 | 25.7 $\pm$ 10.0 | 30.7 $\pm$ 6.2 | 906 | 12.5 | 108.6
90482 Orcus * | Rbio 2.6 | 8 | 38.3 $\pm$ 1.3 | 19.6 $\pm$ 1.6 | 958 | 15.9 | 22.5
90568 2004 GV9 * | Rbio 2.8 | 7 | 9.1 $\pm$ 0.2 | 26.2 $\pm$ 0.5 | 680 | 14.2 | 101.3
95626 2002 GZ32 + | Rbio 2.4 | 7 | 8.27 $\pm$ 1.25 | 4.87 $\pm$ 1.38 | 237 | 11.5 | 433.8
119951 2002 KX14 * | Rbio 3.7 | 8 | 7.9 $\pm$ 0.1 | 34.9 $\pm$ 0.7 | 455 | 7.0 | 290.5
120061 2003 CO1 + | Rbio 2.8 | 7 | 5.16 $\pm$ 0.33 | 0.42 $\pm$ 0.21 | 94 | 29.9 | 52.3
120132 2003 FY128 * | Sym 13 | 8 | 28.0 $\pm$ 6.0 | 10.8 $\pm$ 13.7 | 460 | 3.0 | 18.0
120348 2004 TY364 + | Bior 5.5 | 8 | 2.27 $\pm$ 0.02 | 0.60 $\pm$ 0.01 | 512 | 16.0 | 146.8
134340 Pluto | Bior 3.1 | 6 | 4.31 $\pm$ 0.12 | 76.0 $\pm$ 17.6 | 2372 | 46.1 | 46.2
136108 Haumea ** | Rbio 3.9 | 5 | 59.8 $\pm$ 1.8 | 27.0 $\pm$ 2.6 | 1240 | 13.0 | 14.7
136199 Eris * | Bior 5.5 | 6 | 11.6 $\pm$ 0.8 | 92.4 $\pm$ 11.6 | 2326 | 18.46 | 21.2
144897 2004 UX10 + | Rbio 2.8 | 6 | 18.5 $\pm$ 3.3 | 34.6 $\pm$ 6.9 | 398 | 11.4 | 63.6
145451 2005 RM43 + | Bior 5.5 | 8 | 21.9 $\pm$ 1.8 | 22.0 $\pm$ 1.9 | 252 | 5.4 | 36.1
145452 2005 RN43 * | Bior 5.5 | 8 | 1.08 $\pm$ 0.06 | 0.94 $\pm$ 0.01 | 679 | 18.8 | 126.5
145453 2005 RR43 + | Bior 5.5 | 8 | 76.4 $\pm$ 2.6 | 17.6 $\pm$ 0.3 | 670 | 3.6 | 5.2
174567 Varda + | Db 20 | 8 | 7.0 $\pm$ 0.5 | 9.9 $\pm$ 1.5 | 792 | 12.5 | 128.5
208996 2003 AZ84 +,* | Db 4 | 8 | 20.2 $\pm$ 3.3 | 16.1 $\pm$ 6.7 | 727 | 7.7 | 41.0
229762 2007 UK126 + | Rbio 2.6 | 8 | 11.9 $\pm$ 1.1 | 10.8 $\pm$ 3.1 | 599 | 10.5 | 95.1
250112 2002 KY14 + | Bior 5.5 | 8 | 5.0 $\pm$ 0.3 | 7.9 $\pm$ 0.7 | 47 | 19.8 | 110.1
281371 2008 FC76 + | Bior 5.5 | 8 | 2.88 $\pm$ 0.02 | 13.3 $\pm$ 2.8 | 68 | 10.9 | 231.7
307616 2003 QW90 * | Bior 5.5 | 8 | 3.85 $\pm$ 0.05 | 12.2 $\pm$ 1.3 | 366 | 2.05 | 11.0
309737 2008 SJ236 + | Bior 5.5 | 7 | 12.5 $\pm$ 0.3 | 63.0 $\pm$ 52.5 | 18 | 5.6 | 71.8
455502 2003 UZ413 + | Bior 5.5 | 8 | 25.0 $\pm$ 2.2 | 2.6 $\pm$ 1.8 | 1214 | 7.0 | 130.7
2007 UM126 + | Rbio 3.7 | 7 | 7.4 $\pm$ 0.2 | 19.4 $\pm$ 6.4 | 42 | 10.4 | 130.0
2007 VH305 + | Rbio 2.8 | 6 | 13.2 $\pm$ 4.6 | 23.9 $\pm$ 13.8 | 24 | 5.6 | 39.4
Charon | Rbio 2.8 | 8 | 53.4 $\pm$ 0.7 | 3.2 $\pm$ 0.1 | 1212 | 9.8 | 9.8
Triton222Its believed that Triton had its origin in the trans-Neptunian region, and that it was captured by Neptune during the evolution of the solar system. | Bior 4.4 | 6 | 16.6 $\pm$ 1.1 | 42.4 $\pm$ 3.0 | 2706 | 51.4 | 54.0
Table 1: Spectral parameters of observed objects. This table give us an idea
about the presence of methanol and/or water ices on the surface of theses
objects. We took advantage of the last two columns to show the signal-to-noise
ratio before and after the filtering, and in Col. 2 we show which wavelet was
used to perform the filtering, where Bior is the wavelet Biorthogonal, Rbio is
Reverse biorthogonal, Sym is Symlets, and Db is Daubechies.

##  5 Results

###  5.1 Filtering

We show here two examples that are representative of our sample: a large and a
small body (according to the classification of Brown, 2012).

Figure 3 shows the filtering applied to the spectrum of 136108 Haumea, which
is a large TNO with an apparent brightness high enough to generate a spectrum
with a high signal-to-noise ratio. According to the algorithm, the wavelet
that best filters this spectrum is the wavelet Reverse Bi-orthogonal 3.9 in
the scale 5. The technique was successful in filtering the spectrum because
the positions of the absorption bands of the original spectrum were not
displaced nor were their shapes altered. Analyzing the residue distribution,
we see that there are no structures or patterns, and the signal-to-noise ratio
increased. Zooming into the region between 1.9 and 2.3 $\mu$m (Fig. 4) where
there is absorption due to water ice, it is possible to perceive with more
clarity that the technique did not alter the shape of the band. This indicates
that the wavelets only removed noise while respecting the signal of the
spectrum.

Figure 3: Spectrum of 136108 Haumea. The original data is shown in red, the
filtered spectrum with the wavelet RBio 3.9 (scale 5) is shown in black, and
the difference between the original and filtered spectra is shown in blue.
Figure 4: A zoom-in of the region between 1.9 and 2.3 $\mu$m of the Haumea
spectrum for better visualization of the filtration. The original spectrum
(before filtering) is shown in red, the filtered spectrum with the wavelet
RBio 3.9 (scale 5) is shown in black, and the difference between the original
and filtered spectra is shown in blue.

Figure 5 shows how the technique works on the spectra of smaller TNOs, for
instance the spectrum of 2003 QW90. In this case the optimal scale for
filtering according to the algorithm is the Bi-orthogonal 5.5 in scale 8.
Analyzing Fig. 5, we see that although the amount of noise is large, the
technique did not insert any obvious artifact when trying to filter the
spectrum. The amount of noise decreased, but some noise still remains in the
spectrum. We conclude that the wavelet simple thresholding has a limit: it
filters to the point where it is no longer possible to remove noise without
removing signal from the spectrum. Analyzing the bottom box, we see that the
residue is larger than in the previous case, but there are still no obvious
structures or patterns in it, as can be seen in Fig. 6 where we show a zoom of
the final region of the spectrum of this TNO.

Figure 5: Spectrum of 307616 2003 QW90. The original data is shown in red, the
filtered spectrum with the wavelet Bior 5.5 (scale 8) is shown in black, and
the difference between the original and filtered spectra is shown in blue. The
residue distribution was shifted downward by 1 unit for better visualization.
Figure 6: Zoom-in of the region between 1.9 and 2.3 $\mu$m of TNO 2003 QW90.
The original data is shown in red, the filtered spectrum with the wavelet Bior
5.5 (scale 8) is shown in black, and the difference between the original and
filtered spectra is shown in blue. The residue distribution was shifted
downward by 1 unit for better visualization.

###  5.2 Limitations

It can be seen that, whenever the spectra have high S/No, the absorption
structures are very well respected. In the case of lower S/No this might not
strictly be the case because the wavelets’ optimal performance assume a signal
that is contaminated by white (Gaussian) noise.

Our data was obtained by photon counting, and is thus contaminated by Poisson
noise. This is not a big problem since in the limit of large numbers the
Poisson distribution tends to the Gaussian (see Bevington and Robinson, 2003).
More complicated are systematic noises (instrumental and calibration), which
do not necessarily follow a statistical distribution and will affect any
filtering technique. Examples of this can be seen in Figs. 1 and 2 with a
feature close to 2.2 $\mu$m due to incomplete removal of telluric features.

In summary, through this analysis we conclude that the application of the
wavelets technique worked well to filter out TNO spectra in both cases (low
and high signal-to-noise ratios); therefore, we will assume it works just as
well for the intermediate cases. Now, we show the results of the comparisons
between these de-noised spectra and the spectra of water and methanol ices in
order to find evidence of these ices on the surfaces of these objects.

###  5.3 Comparison

####  5.3.1 Water ice

According to Table 1 the candidates in our sample that possess water ice are
Quaoar, Orcus, 2003 FY128, Haumea, Sedna, 2005 RR43, and Charon (all with
D${}_{2.0}>$ 25%). With shallower bands we have 2003 AZ84, 2005 RM43, 1996
TL66, 2004 UX10, 2003 UZ413, Triton, Crantor, and Huya (15 $<$ D${}_{2.0}<$
25%). Among these, those that showed possible evidence through direct
comparison with the water ice spectrum were Quaoar, Orcus, Haumea, 2005 RR43,
and Charon. The amount of noise remaining in the spectra of the other TNOs did
not allow the identification of the possible water ice band. However, the drop
in flux in the 2 $\mu$m region is indicative that some absorption is occurring
in the region. Figure 7 shows spectra of objects displaying clear water ice
features.

For large TNOs we note the similarity between the water ice spectrum and the
object spectra. The band at 1.65 $\mu$m is conspicuous in all spectra. This
band indicates the presence of crystalline structure ice on the surface of
TNOs, which might be related to a recent exposition of the ice that has had no
time to become amorphous due to space weathering. The band at 2 $\mu$m appears
in all spectra, but with different depths.

Figure 7: Normalized reflectance vs. wavelength. The spectra were shifted in
flux for better visualization. The region close to 1.85 $\mu$m was masked due
to the absorptions of the atmosphere. In green the spectrum of Charon, in blue
Haumea, in red 2005 RR43, in purple Orcus, and in brown, Quaoar. There is a
coincidence of the band at 1.65 and 2 $\mu$m in all spectra of TNOs, which are
signatures of the water ice in these objects.

####  5.3.2 Methanol ice

The objects in our sample with evidence of having methanol ice, according to
the criterion proposed in Merlin et al., (2012) and after comparison with the
spectrum obtained in laboratory to methanol ice, are 2004 TY364, 2003 QW90,
Pholus, Huya, Crantor, and 2005 RR43. For the other objects in our sample it
was not possible to find structures that could indicate the presence of this
ice. In Fig. 8 we show the larger objects in the sample, while in Fig. 9 we
show the smaller objects with possible presence of methanol in their surface
composition.

Our analysis focuses on the final part of the spectrum, specifically the
region beyond 2 $\mu$m. In this region the amount of noise is large and
therefore is it difficult to detect methanol ice structures for any size
range. However, it is possible to perceive signs as flux drop that could be
associated with small and shallow absorption features. Nevertheless, care must
be taken not to misinterpret some spurious features, as mentioned above.

Figure 8: Normalized flux vs. wavelength. The spectra were shifted in flux for
better visualization. From top to bottom the spectra shown are 2004 TY364 in
blue, the model of methanol (CH3OH) in black, and 2005 RR43 in red. Figure 9:
Normalized flux vs. wavelength. The spectra were shifted in flux for better
visualization. From top to bottom, the spectra shown are Crantor in brown,
Huya in purple, the model of methanol (CH3OH) in black, 2003 QW90 in red, and
Pholus in blue.

##  6 Conclusions

In this paper, we show how the application of the wavelets allowed us to
analyze the spectra of 50 trans-Neptunian objects with diameters between 50
and 2300 km. Some of the objects in our sample already have data available in
the literature and were taken as validators of the technique. As the results
were positive, we applied the technique for the rest of the objects in our
sample.

We performed the filtering of the spectra of the sample in a two-step process.
First we developed an algorithm that determines the best wavelet and the
respective scale that optimizes the result of the filtering, and we filtered
the spectra with the wavelet and scale indicated by the algorithm. After
filtration, we calculated some spectral parameters to find out in which of the
sample spectra there was the possibility of finding structures similar to
those found in water and methanol ice spectra. We compared the filtered
spectra of the candidates with the spectra calculated from optical constants
measured in the laboratory for water and methanol ices. According to the
similarity between the spectra, we were able to recognize the absorption bands
associated with the water ice and to identify some evidence of the presence of
methanol ice in some of the sample spectra. These results are summarized in
Table 2.

We consider that the application of wavelets to filter the spectra was
successful because it was possible to analyze TNO spectra that had previously
been considered unusable due to the amount of noise they had.

Name | Ices
---|---
5145 Pholus | Methanol+
38628 Huya | Methanol*
50000 Quaoar | Water+
83982 Crantor | Methanol+
90482 Orcus | Water+
120348 2004 TY364 | Methanol+
136108 Haumea | Water+
145453 2005 RR43 | Water+, Methanol*
307616 2003 QW90 | Methanol*
Charon | Water+
Table 2: Summary of our results: the TNOs with the respective ices. The ices
marked with + have detections in the literature, and those with * need
confirmation.

We checked whether there was any relation between dynamical grouping
(populations) and the possible existence of ices. We did not find any result
of relevance, due to the small number statistics. Future developments will
have to wait until new data becomes available, especially for smaller objects,
when we will be able to apply filtering by wavelets to extract the maximum
information possible.

Acknowledgements. This work was developed during A.C.S.F.’s M.Sc. degree at
the National Observatory with financial support from CAPES. A.A.C. would like
to acknowledge support from CNPq and FAPERJ. Y.J.T. acknowledges financial
support from the Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado
do Rio de Janeiro - FAPERJ (fellowship Nota 10, PDR-10). This research is
based on observations collected at the European Organisation for Astronomical
Research in the Southern Hemisphere under ESO program 178.C-0867. We thank the
comments and suggestions of the referee and the editor that improved the
quality of this work.

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# INCPrompt: Task-Aware incremental Prompting for Rehearsal-Free Class-
incremental Learning

###### Abstract

This paper introduces INCPrompt, an innovative continual learning solution
that effectively addresses catastrophic forgetting. INCPrompt’s key innovation
lies in its use of adaptive key-learner and task-aware prompts that capture
task-relevant information. This unique combination encapsulates general
knowledge across tasks and encodes task-specific knowledge. Our comprehensive
evaluation across multiple continual learning benchmarks demonstrates
INCPrompt’s superiority over existing algorithms, showing its effectiveness in
mitigating catastrophic forgetting while maintaining high performance. These
results highlight the significant impact of task-aware incremental prompting
on continual learning performance.

Index Terms—  Continual Learning, Catastrophic Forgetting, Prompt Learning,
Incremental Learning

##  1 Introduction

Continual learning, the process of acquiring new tasks without erasing
knowledge of previously learned ones, presents a substantial hurdle in deep
learning. Catastrophic forgetting, the tendency of models to lose knowledge
from previous tasks when learning new ones, is a pressing issue that hinders
the practical application of sizeable pre-trained vision models [1, 2, 3].

In response to these challenges, researchers have proposed various continual
learning strategies broadly categorized into two methods: regularization-based
and rehearsal-based. Regularization-based methods constrain parameter updates
during new task learning to protect old knowledge, typically adding
regularization terms to the loss function [4, 5, 6]. Rehearsal-based methods
maintain some previous task data, mixing them with new task data during
training [7, 8, 9].

(a) Traditional rehearsal-based methods can be computationally expensive and
inefficient due to storing and accessing large amounts of data.

(b) INCPrompt offers a more efficient strategy. It utilizes a prompt-
generating model. This model stores task-specific knowledge, removing the need
for a rehearsal buffer.

Fig. 1: Comparison of traditional methods and INCPrompt.

Prompt learning strategically modifies the input text, providing language
models with insights specific to the task. The design of effective hand-
crafted prompts can be complex, often requiring heuristic approaches. Soft
prompting [10] and prefix tuning [11] circumvent this by introducing learnable
parameters that guide the model’s behavior, simplifying the adaptation of
large language models to specific tasks. There have been substantial
advancements in this field, such as the L2P [12] method, which learns a
mapping from task descriptions to prompts. The DualPrompt [13] approach
utilizes dual prompts to enhance model control. However, these methods lack
our proposed method’s dynamic, task-specific adaptability.

To address these challenges, we advance the field through a series of novel
contributions, detailed below:

  * •

We introduce INCPrompt, which balances new task adaptability and knowledge
retention from previous tasks.

  * •

It incorporates a regularization term to maintain a balance between adaptation
to reduce overfitting and encourage feature extraction.

  * •

We conduct comprehensive experimental evaluations against other methods across
various benchmarks.

##  2 The Proposed Method

###  2.1 Problem Setting

(a) Training procedure

(b) Inference procedure

Fig. 2: The architecture of our algorithms.

A model is exposed to a sequence of $N$ tasks $D={D_{1},...,D_{N}}$ in our
setting. Each task exclusively contains different semantic object categories,
with the objective being to progressively acquire the ability to categorize
newly introduced object classes while preserving the classification accuracy
of object classes learned in the past. Here’s a formalized definition of the
continual learning problem:

For each task $D_{t}$ consisting of pairs $(x_{i,t},y_{i,t})$ with $i$ ranging
from 1 to $n_{t}$, a model $f_{\theta}$, parameterized by $\theta$, aims to
infer the label $y_{i,t}\in Y$ for any input $x_{i,t}\in X$, thereby learning
the function from $X$ to $Y$. We define the task-specific loss
$L_{\text{task}}$ as the cross-entropy loss between the predicted labels and
the true labels:

$L_{\text{task}}=-\sum_{i=1}^{n_{t}}\sum_{c=1}^{|Y|}y_{i,t,c}\log(f_{\theta}(x_{i,t})_{c})$
(1)

where $y_{i,t,c}$ is an indicator variable, 1 if the class label $c$ is the
correct classification for the observation $x_{i,t}$, and 0 otherwise. It
should be emphasized that the data from earlier tasks is inaccessible when
training subsequent tasks.

###  2.2 Key Learner

The Key-Learner implements an attention-based transformation to the input
tokens. The transformed input is then processed by a self-attention mechanism,
which is mathematically represented as:

$\text{Attn}(Q_{i},K_{i},V_{i})=\text{softmax}\left(\frac{Q_{i}K_{i}^{T}}{\sqrt{d_{k_{i}}}}\right)V_{i}$
(2)

In this updated equation, $Q_{i}$, $K_{i}$, and $V_{i}$ represent the query,
key, and value of the output tokens from the Vision Transformer encoder, and
$d_{k_{i}}$ is the dimensionality of the queries and keys. The subscript $i$
denotes that these are specific output tokens from the ViT.

This triplet loss function maximizes the softmax probability of a vector’s dot
product with a positive sample compared to negative samples. It’s defined as:

$L_{\text{Triplet}}=\sum_{i=1}^{N}[|T_{k}^{i}-K_{i}^{a}|_{2}^{2}-|T_{k}^{i}-K_{i}^{n}|_{2}^{2}+\alpha]+$
(3)

In this equation, $T_{k}^{i}$ represents the output tokens for a given input.
$K_{i}^{a}$ represents the key output from the current key-learner as an
anchor point. $K_{i}^{n}$ represents the key output from a different key-
learner with the highest similarity to the output tokens $T_{k}^{i}$ among all
other key-learners. $|\cdot|_{2}$ denotes the Euclidean distance, $[\cdot]+$
denotes the operation of taking the maximum between the argument and 0, and
$\alpha$ is a margin parameter that is used to separate positive and negative
pairs, which can be tuned based on the specific task.

This technique prevents overfitting and encourages feature extraction with a
penalty to the $L_{1}$ norm of the key. It’s defined as:

$L_{\text{reg}}=\sum_{i=1}^{N}||K_{i}^{a}||_{1}$ (4)

In this equation, $||K_{i}^{a}||_{1}$ denotes the L1 norm of the key output
from the current key-learner. A higher regularization loss will be obtained if
the two have a lower difference. This term encourages the model to keep the
$K_{P}$ different from output tokens and thus helps to prevent overfitting.
The total loss of key-learner is denoted as:

$L_{key}=\lambda_{\text{reg}}\cdot L_{reg}+L_{Triplet}$ (5)

$\lambda_{\text{reg}}$ is a hyper-parameter and serves as a regularization
coefficient that balances the L1-norm regularization term $L_{reg}$ and the
triplet loss $L_{Triplet}$.

###  2.3 Task-aware Prompter

Moreover, we utilize an ensemble of feedforward layers with a non-linear
activation function, termed the ‘prompter,’ to dynamically synthesize the key
($P_{k}$) and value ($P_{v}$) components of the prompt $P$. The generation of
the prompt by the prompter can be represented by the following equation:

$P=f_{\text{prompter}}(x_{\text{token}})$ (6)

Here, $f_{\text{prompter}}$ corresponds to the prompter function, which
receives an image token as input.

In our proposed INCPrompt method, we integrate an external prompt $P$ into the
computation process. The prompt $P$ consists of a key component $P_{k}$ and a
value component $P_{v}$. The modified attention computation becomes:

$Att(Q,K\oplus P_{k},V\oplus P_{v})=f(\frac{Q(K\oplus
P_{k})^{T}}{\sqrt{d_{k}}})(V\oplus P_{v})$ (7)

where $f(.)$ denotes the softmax function, $\oplus$ denotes the concatenation
operation.

In this way, the transformer model can consider the original input and the
external prompt when computing the attention. This mechanism ensures that the
generated prompts are adaptive and can effectively incorporate task-relevant
information into the attention mechanism of the transformer model.

###  2.4 Algorithm Architecture

Algorithm 1 Task-Aware Incremental Prompting

1:Input: Tokens from ViT $x_{token}$, Task ID $x_{id}$

2:Output: Final prompter (p), Contrastive loss (L) if in training mode

3:Initialize: Pre-trained Vision Transformer (V), Key-Learners (K), Task-
Prompters (T), Image dataset (D),

4:if Train then

5: $K_{p},K_{n}\leftarrow K(x_{id},x_{token})$ $\triangleright$ Generate pos.
and neg.keys

6: $L\leftarrow L_{key}(K_{p},K_{n},t)$ $\triangleright$ Compute key loss

7: $P\leftarrow T(x_{id},x_{token})$ $\triangleright$ Get Task-prompter output

8:end if

9:if Eval then

10: $K_{All}\leftarrow K(x_{token})$ $\triangleright$ Generate all keys

11: $idx\leftarrow argmax(Sim(K_{All},x_{token}))$ $\triangleright$ Find the
best matching K

12: $P\leftarrow T(x_{id},x_{token})$ $\triangleright$ Get the output of the
Prompter

13:end if

14:$p\leftarrow Divide(P)$ $\triangleright$ Divide the prompts into key and
value

15:if Train then

16: return $p,l$

17:else

18: return $p,0$

19:end if

Consequently, the loss function of the INCPrompt is computed as the sum of the
components mentioned above:

$L=L_{task}+L_{key}$ (8)

By minimizing this loss function, INCPrompt can effectively learn task-
specific and retain task-invariant knowledge, improving performance in
continual learning tasks, as shown in Algorithm 1.

##  3 evaluation

###  3.1 Experiment Setup

Fig. 3: Histograms depicting prompter selection for (left) Split CIFAR-100 and
(right) ImageNet-R. ImageNet-R has a higher level of intra-task similarity
compared to CIFAR100. On the other hand, CIFAR100 benefits more from prompts
that are more specific to individual tasks. For better readability, we only
display the first five tasks.

To illustrate our primary findings, we employ the Split CIFAR-100 and Split
ImageNet-R benchmark, an adaptation of ImageNet-R [14], comprising 200 classes
randomly divided into ten tasks, each containing 20 types. It is sufficiently
robust to expose a high forgetting rate in advanced CL methods during class-
incremental learning in these two datasets.

As shown in Table 1, to ensure a thorough and unbiased experiment, we
initially compare our proposed method, INCPrompt, with techniques based on
regulation, rehearsal, and prompting approaches. We have chosen characteristic
methods like EWC [5], LwF [4], ER [15], GDumb [16], BiC [17], DER++ [18], Co2L
[19], L2P [12], and DualPrompt [13], from all categories. To provide a more
transparent illustration, we include FT-seq, representing a straightforward
sequential training method, and Upper-bound, representing traditional fine-
tuning encompassing all tasks. Notably, the accuracy figures reported for our
INCPrompt methodology are derived under the specific conditions of employing a
prompt length of 20 and a prompt depth of 6.

Table 1: Outcomes in continual learning (where task ID remains unavailable during testing). The methods are compared and grouped according to their buffer sizes. A buffer size of zero indicates that no rehearsal was utilized, rendering most state-of-the-art techniques inapplicable. Method | Dataset Buffer | Split CIFAR-100 | Dataset Buffer | Split ImageNet-R
---|---|---|---|---
|  | Avg. Acc (↑) | Forgetting (↓) |  | Avg. Acc (↑) | Forgetting (↓)
ER [15] | 1000 | 67.87±0.57 | 33.33±1.28 | 1000 | 55.13±1.29 | 35.38±0.52
BiC [17] | 1000 | 66.11±1.76 | 35.24±1.64 | 1000 | 52.14±1.08 | 36.70±1.05
GDumb [16] | 1000 | 67.14±0.37 | - | 1000 | 38.32±0.55 | -
DER++ [18] | 1000 | 61.06±0.87 | 39.87±0.99 | 1000 | 55.47±1.31 | 34.64±1.50
Co2L [19] | 1000 | 72.15±1.32 | 28.55±1.56 | 1000 | 53.45±1.55 | 37.30±1.81
ER [15] | 5000 | 82.53±0.17 | 16.46±0.25 | 5000 | 65.18±0.40 | 23.31±0.89
BiC [17] | 5000 | 81.42±0.85 | 17.31±1.02 | 5000 | 64.63±1.27 | 22.25±1.73
GDumb [16] | 5000 | 81.67±0.02 | - | 5000 | 65.90±0.28 | -
DER++ [18] | 5000 | 83.94±0.34 | 14.55±0.73 | 5000 | 66.73±0.87 | 20.67±1.24
Co2L [19] | 5000 | 82.49±0.89 | 17.48±1.80 | 5000 | 65.90±0.14 | 23.36±0.71
FT-seq | 0 | 33.61±0.85 | 86.87±0.20 | 0 | 28.87±1.36 | 63.80±1.50
EWC [5] | 0 | 47.01±0.29 | 33.27±1.17 | 0 | 35.00±0.43 | 56.16±0.88
LwF [4] | 0 | 60.69±0.63 | 27.77±2.17 | 0 | 38.54±1.23 | 52.37±0.64
L2P [12] | 0 | 83.86±0.28 | 7.35±0.38 | 0 | 61.57±0.66 | 9.73±0.47
DualPrompt [13] | 0 | 84.01±0.28 | 7.01±0.21 | 0 | 71.13±0.42 | 6.79±0.12
Ours | 0 | 85.03±0.21 | 6.64±0.18 | 0 | 73.22±0.45 | 6.47±0.16
Upper-bound | - | 90.85±0.12 | - | - | 79.13±0.18 | -

###  3.2 Results Analysis

Table 1 summarizes the methods compared on the two datasets. Our proposed
approach, INCPrompt, consistently outperforms all competing methods, including
those that do not involve rehearsal and those that rely on rehearsal. Notably,
when the buffer size is set to 5000, all rehearsal-based methods exhibit
performance similar to GDumb’s. This observation suggests that the
implementation of rehearsal-based continual learning may not offer substantial
performance improvement over supervised training. Figure 5 shows that prompts
added to the 5th MSA layers yield the most effective results. Figure 5 aims to
highlight that, as prompt length increases, accuracy reaches a saturation
point.

Fig. 4: The relationship between the accuracy and the prompt layer depth of
ViT blocks. (prompt length=20)

Fig. 5: The relationship between the accuracy and the length of the prompt.
(layer depth=5)

##  4 Conclusion

This work introduces INCPrompt, a unique continual learning approach
leveraging dynamic, task-specific prompts to counter catastrophic forgetting.
This innovative blend of key-learners and task-aware prompters enables
INCPrompt to maintain general and task-specific knowledge. Testing on various
continual learning datasets confirms its balance between new task adaptability
and old task knowledge preservation.

##  5 ACKNOWLEDGEMENT

Sponsored by the Key Research and Development Program of Guangdong Province
under grant No.2021B0101400003, Tsinghua-Toyota Joint Research Fund.

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