# Seasonal Contrast:
Unsupervised Pre-Training from Uncurated Remote Sensing Data

Oscar Mañas1,2 Alexandre Lacoste1 Xavier Giró-i-Nieto2 David Vazquez1 Pau
Rodriguez1
1Element AI 2Universitat Politècnica de Catalunya
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Remote sensing and automatic earth monitoring are key to solve global-scale
challenges such as disaster prevention, land use monitoring, or tackling
climate change. Although there exist vast amounts of remote sensing data, most
of it remains unlabeled and thus inaccessible for supervised learning
algorithms. Transfer learning approaches can reduce the data requirements of
deep learning algorithms. However, most of these methods are pre-trained on
ImageNet and their generalization to remote sensing imagery is not guaranteed
due to the domain gap. In this work, we propose Seasonal Contrast (SeCo), an
effective pipeline to leverage unlabeled data for in-domain pre-training of
remote sensing representations. The SeCo pipeline is composed of two parts.
First, a principled procedure to gather large-scale, unlabeled and uncurated
remote sensing datasets containing images from multiple Earth locations at
different timestamps. Second, a self-supervised algorithm that takes advantage
of time and position invariance to learn transferable representations for
remote sensing applications. We empirically show that models trained with SeCo
achieve better performance than their ImageNet pre-trained counterparts and
state-of-the-art self-supervised learning methods on multiple downstream
tasks. The datasets and models in SeCo will be made public to facilitate
transfer learning and enable rapid progress in remote sensing
applications.111Code, datasets and pre-trained models are available at
https://github.com/ElementAI/seasonal-contrast

##  1 Introduction

Remote sensing is becoming increasingly important to many applications,
including land use monitoring [12], precision agriculture [29], disaster
prevention [37], wildfire detection [11], vector-borne disease surveillance
[20], and tackling climate change [33]. Combined with recent advances in deep
learning and computer vision, there is enormous potential for monitoring
global issues through the automated analysis of remote sensing and other
geospatial data streams.

Remote sensing provides a vast supply of data. The number of Earth-observing
satellites is continuously growing, with over 700 satellites currently in
orbit generating terabytes of imagery data every day [30]. However, many
downstream tasks of interest are constrained by a lack of annotations, which
are particularly costly to obtain since they often require expert knowledge,
or expensive ground sensors. In recent years, a number of techniques have been
developed to mitigate the need for labeled data [24, 26, 25], but their
application to remote sensing images is largely underexplored.

Furthermore, existing remote sensing datasets [38, 19, 42] are highly curated
to form well-balanced and diversified classes. Simply discarding the labels
does not undo this careful selection of examples, which also requires
considerable human effort. Our goal is to exploit the massive amount of
publicly available remote sensing data for learning good visual
representations in a truly unsupervised way. To enable this, we construct a
remote sensing dataset from Sentinel-2 [10] tiles without any human
supervision, neither for curating nor annotating the data.

Figure 1: Distribution of the Seasonal Contrast (SeCo) dataset. Each point
represents a sampled location. Images are collected around human settlements
to avoid monotonous areas such as oceans and deserts.

Another characteristic unique to remote sensing data is satellite revisit,
which describes the ability of the system to make repeated image captures of
the same point of the Earth’s surface over time. For publicly funded satellite
constellations such as Sentinel [10] or Landsat [35], the revisit time is of
the order of days. This temporal dimension provides an additional source of
natural variation which complements the artificial augmentation of images. For
instance, no amount of artificial augmentation can show how a snowy mountain
summit looks like when the snow melts down, or how the different stages of a
crop change through the seasons.

Self-supervised learning methods have recently emerged as an effective
methodology to learn from vast amounts of unlabeled data. Contrastive methods
push together representations of images that are semantically similar (i.e.
positive pairs). Since no labels are available, traditional contrastive
learning methods that work with natural images use different artificial
augmentations of the same image (views) as positive pairs. In the case of
remote sensing images, we propose to leverage the temporal information to
obtain pairs of images from the same location at different points in time,
which we call seasonal positive pairs. We argue that seasonal changes provide
more semantically meaningful content than artificial transformations, and
remote sensing images provide this natural augmentation for free.

We propose Seasonal Contrast (SeCo), a novel methodology for pre-training
rich, transferable representations for remote sensing applications. SeCo
consists of two parts, an unsupervised data acquisition procedure and a self-
supervised learning model to learn from the acquired data. The self-supervised
learning model is designed based on the observation that encouraging the
representation to be invariant to seasonal changes is a strong inductive bias.
This property can be beneficial for certain downstream tasks where the
prediction will not change with seasonal variations (e.g. land-cover
classification, agricultural pattern segmentation, building detection), but
harmful for downstream tasks where seasonal variations are important (e.g.
deforestation tracking, change detection). We would like to learn good
representations of remote sensing images that are agnostic to the downstream
tasks where they could be applied.

To leverage temporal information without limiting the visual representations
to be always invariant to time, we use the idea of multiple embedding sub-
spaces [47]. Instead of mapping an image to a single embedding space which is
invariant to all augmentations, we construct separate embedding sub-spaces and
optimize them to be variant or invariant to seasonal changes. We use a multi-
head architecture with a shared backbone which produces a common
representation that encodes the different variances and invariances. Once the
model is trained, this representation can be applied to a wide range of remote
sensing downstream tasks, where the model can selectively utilize the
different factors of variation captured in the representation.

We evaluate SeCo on several remote sensing datasets and tasks. Our experiments
on land-cover classification with BigEarthNet [38] and EuroSAT [19], and
change detection with OSCD [8] demonstrate that SeCo pre-training is more
effective for remote sensing tasks than the common ImageNet [36] and MoCo [18]
pre-training.

In summary, our contributions are:

  * •

We describe a general method for collecting uncurated and unlabeled datasets
of remote sensing images. We use this method to construct a remote sensing
dataset from Sentinel-2 tiles without any human supervision.

  * •

We combine recent contrastive self-supervised learning methods with the
temporal information provided by satellites to learn good visual
representations which are simultaneously variant and invariant to seasonal
changes.

  * •

We obtain state-of-the-art results on BigEarthNet and EuroSAT land-cover
classification, and on OSCD change detection.

##  2 Background

Self-supervised learning is the branch of unsupervised learning where the data
itself provides the supervision. The main idea is to occlude or perturb part
of the data and task the network with predicting it from the visible data.
This defines a pretext task (or proxy loss) and the network is forced to learn
what we care about the data (e.g. a semantic representation) in order to solve
it. A variety of pretext tasks have been proposed for images, such as
predicting the relative position of patches [9], solving jigsaw puzzles [31],
predicting rotations [13] or colorization [48].

More recently, contrastive pretext tasks [46, 32, 41, 18, 28, 5, 16, 4] have
dominated the subfield of self-supervised learning, demonstrating superior
performance in various downstream tasks. Intuitively, contrastive learning
methods pull together the representations of similar examples while pushing
apart the representations of dissimilar examples. Since the examples are not
labeled, these methods make the assumption that each example defines and
belongs to its own class. Hence, positive pairs are generated by applying
random augmentations to the same example, while negative pairs come from other
instances in the dataset.

Formally, this task can be formulated as a dictionary look-up problem, where a
given example $x$ is augmented into two views, query $x^{q}$ and key $x^{k}$,
an encoder network $f$ maps the examples into an embedding space, and the
representation of the query $q=f(x^{q})$ should be closer to the
representation of its designated key $k^{+}=f(x^{k})$ than to the
representation of any negative key $k^{-}$ coming from a set of randomly
sampled instances different from $x$. To this end, a contrastive objective is
optimized over a batch of positive/negative pairs. A common choice is the
InfoNCE loss [32]:

$\mathcal{L}=-\mathrm{log}\frac{\mathrm{exp}(q\cdot
k^{+}/\tau)}{\mathrm{exp}(q\cdot k^{+}/\tau)+\sum_{k^{-}}\mathrm{exp}(q\cdot
k^{-}/\tau)}$ (1)

where $\tau$ is a temperature hyper-parameter scaling the distribution of
distances.

##  3 Method

We propose a methodology for pre-training rich, transferable representations
for remote sensing imagery, consisting of a general procedure for collecting
an unsupervised pre-training dataset (Section 3.1) and a self-supervised
learning method (Section 3.2) for leveraging this data.

Figure 2: Diagram of the Seasonal Contrast method. A query image ($q$) is
augmented with temporal ($k_{0},k_{1}$) and synthetic ($k_{0},k_{2}$)
transformations $\mathcal{T}$. Image embeddings produced by the encoder $f$
are projected into three different sub-spaces by heads $h_{0},h_{1},h_{2}$.
Green boxes represent positive pairs while red boxes represent negative pairs
(i.e. including images from other locations). Sub-space $\mathcal{Z}_{0}$ is
invariant to all transformations, thus all keys belong to the same class as
the query. $\mathcal{Z}_{1}$ is invariant to seasonal augmentations, while
$\mathcal{Z}_{2}$ is invariant to synthetic augmentations.

###  3.1 Unsupervised Dataset Collection

Remote sensing provides a vast amount of imagery data, but annotations are
usually scarce, and domain expertise or ground sensors are often required
[21]. In order to train on a large amount of satellite images, we collect a
new dataset of Sentinel-2 [10] patches without any human supervision.

The Sentinel-2 imagery consists of 12 spectral bands (including RGB and NIR)
at 10 m, 20 m and 60 m resolution, with a revisit time of around 5 days. We
use Google Earth Engine [15] to process and download image patches from about
200K locations around the world, where each patch covers a region of roughly
$2.65\times 2.65$ km. At each location, we download 5 images from different
dates separated by approximately 3 months, which capture the seasonal changes
that occurred in the region over a year. To avoid getting images from the same
periods of the year, at each location we jitter the dates for up to a year. We
also filter out Sentinel-2 tiles with a cloud percentage higher than 10%. In
total, we obtain about 1 million multi-spectral image patches, which amount to
a total of over 387 billion pixels.

##### Sampling Strategy

Our objective is to learn an encoder that can be used on a wide variety of
downstream tasks. To this end, we need to sample from a wide variety of
regions on the Earth. Uniform sampling would lead to a large amount of
redundancy in the types of images. For example, oceans cover 71% of the
planet, forests cover 31% of land, and deserts cover 33% of land. To work
around this, we make the assumption that most of the variability can be
observed in the greater areas around cities. The cities themselves contain a
wide range of constructions, a few kilometers away from cities we often
observe a variety of crops and industrial facilities. Finally, in the range of
50 km-100 km away from cities, we usually observe natural environments. Hence,
we sample around cities following this heuristic (see results in Figure 1):

  1. 1.

Sample uniformly among the 10k most populated cities, and then sample a set of
coordinates from the Gaussian distribution spanning a standard deviation of 50
km around the center of the city.

  2. 2.

Randomly select a reference date over the past year. Add periodic increments
of 3 months to obtain the sampling dates.

  3. 3.

For a 15-day range around each date, check if there exists a Sentinel-2 tile
with less than 10% of cloud coverage that intersects with the coordinates.

  4. 4.

If there exists a valid Sentinel-2 tile for this location on all dates,
process and download all the image patches. Otherwise, go to step 1.

We do not perform any additional data cleaning to ensure that the obtained
images are diverse, informative and free of clouds. Because our dataset is
constructed automatically, we can easily gather more data (more locations,
more dates per location). In this work, however, we limit the scale to a total
of 1M images to make it more comparable to ImageNet [36].

###  3.2 Seasonal Contrast

Given an unsupervised dataset of remote sensing images with temporal
information, we learn a representation that takes advantage of the structure
of the data. We get inspiration from [47] to develop a multi-augmentation
contrastive learning method. This approach can selectively prevent information
loss incurred by artificial augmentations, and extend it with natural
augmentations provided by the seasonal changes on remote sensing images.
Instead of projecting every view to a common embedding space which is
invariant to all augmentations, a common representation is projected into
several embedding sub-spaces which are variant or invariant to time (see
Figure 2). Hence, the shared representation will contain both time-varying and
invariant features, which will transfer efficiently to remote sensing
downstream regardless of whether they involve temporal variation.

####  3.2.1 Views Generation

Given a reference image (query), we produce multiple positive pairs (keys)
with seasonal and artificial augmentations. Let $\mathcal{T}$ be a set of
commonly used artificial augmentations [18], such as random cropping, color
jittering, and random flipping. We first obtain 3 images from the same
location at different times, $x^{t_{0}}$, $x^{t_{1}}$ and $x^{t_{2}}$, which
are randomly selected among all the available ones for that location. No
additional transformations are applied to the query image, i.e.
$x^{q}=x^{t_{0}}$. Hence, $x^{t_{1}}$ and $x^{t_{2}}$ can be considered
seasonal augmentations (or temporal views) of $x^{q}$. The first key view
contains both seasonal and artificial transformations,
$x^{k_{0}}=\mathcal{T}(x^{t_{1}})$, the second key view contains only seasonal
augmentations, $x^{k_{1}}=x^{t_{2}}$, and the third view contains only
artificial augmentations, $x^{k_{2}}=\mathcal{T}(x^{t_{0}})$.

####  3.2.2 Multiple Embedding Sub-spaces

The query and key views are encoded by a neural network $f$ into
representations $v^{q}$, $v^{k_{0}}$, $v^{k_{1}}$, $v^{k_{2}}$ in a common
embedding space $\mathcal{V}\in\mathbb{R}^{d}$. Next, each intermediate
representation is projected into 3 different sub-spaces $\mathcal{Z}_{0}$,
$\mathcal{Z}_{1}$, $\mathcal{Z}_{2}\in\mathbb{R}^{d^{\prime}}$ by non-linear
projection heads $h_{0}$, $h_{1}$, $h_{2}$, where
$h_{i}:\mathcal{V}\mapsto\mathcal{Z}_{i}$. Following recent literature [44],
the embedding sub-spaces are $l_{2}$-normalized, effectively restricting them
to the unit hypersphere.

The embedding sub-space $\mathcal{Z}_{0}$ is invariant to all augmentations,
$\mathcal{Z}_{1}$ is invariant to seasonal augmentations but variant to
artificial augmentations, and $\mathcal{Z}_{2}$ is invariant to artificial
augmentations but variant to seasonal augmentations. Namely, in
$\mathcal{Z}_{0}$ all embeddings $z_{0}^{i}$ should be pulled together, in
$\mathcal{Z}_{1}$ only $z_{1}^{q}$ and $z_{1}^{k_{1}}$ should be pulled
together and pushed apart from $z_{1}^{k_{0}}$ and $z_{1}^{k_{2}}$, and in
$\mathcal{Z}_{2}$ only $z_{2}^{q}$ and $z_{2}^{k_{2}}$ should be pulled
together and pushed apart from $z_{2}^{k_{0}}$ and $z_{2}^{k_{1}}$. This is
represented visually in Figure 2.

A contrastive learning objective is optimized on each embedding sub-space
based on Equation 1, where the definition of positive (and negative) pairs
depends on the invariances (and variances) that are encoded. In
$\mathcal{Z}_{0}$, the positive pair for the query $z_{0}^{q}$ is
$z_{0}^{k_{0}}$, and the negative pairs are embeddings of other instances in
this embedding sub-space. For embedding sub-space $\mathcal{Z}_{1}$, the
positive pair for the query $z_{1}^{q}$ is $z_{1}^{k_{1}}$, while the negative
pairs are embeddings of other instances in this embedding sub-space, plus
$z_{1}^{k_{0}}$ and $z_{1}^{k_{2}}$. Note that $z_{1}^{k_{0}}$ and
$z_{1}^{k_{2}}$ are harder negative pairs for $z_{1}^{q}$ as they come from
the same instance but have a different artificial augmentation. Positive and
negative pairs in embedding space $\mathcal{Z}_{2}$ are analogous to
$\mathcal{Z}_{1}$.

The final learning objective is the sum of all the embedding sub-space losses.
The encoder $f$ must preserve time-varying and invariant information in the
general embedding space $\mathcal{V}$ in order to optimize the combined
contrastive learning objectives of all normalized embedding sub-spaces
$\mathcal{Z}_{i}$. Note that the original contrastive learning objective [32]
is a particular case of multi-augmentation contrastive learning when only the
embedding sub-space $\mathcal{Z}_{0}$ is used.

The representation for transfer learning is taken from the general embedding
space $\mathcal{V}$, since we do not assume any a priori knowledge about the
downstream tasks. In case the right invariances for downstream tasks were
known, the representation could be extracted from a particular embedding sub-
space $\mathcal{Z}_{i}$.

Pre-training | Backbone | 100k images | 1M images
---|---|---|---
Linear probing | Fine-tuning | Linear probing | Fine-tuning
10% | 100% | 10% | 100% | 10% | 100% | 10% | 100%
Random init. | ResNet-18 | 43.05 | 45.95 | 68.11 | 79.80 | 43.05 | 45.95 | 68.11 | 79.80
ImageNet (sup.) | 65.69 | 66.40 | 78.76 | 85.90 | 65.69 | 66.40 | 78.76 | 85.90
MoCo-v2 | ResNet-18 | 69.70 | 70.90 | 78.76 | 85.17 | 69.28 | 70.79 | 78.33 | 85.23
MoCo-v2+TP | 70.20 | 71.08 | 79.80 | 85.71 | 72.58 | 73.60 | 80.68 | 86.59
SeCo (ours) | 74.67 | 75.52 | 81.49 | 87.04 | 76.05 | 77.00 | 81.86 | 87.27
Random init. | ResNet-50 | 43.95 | 46.92 | 69.49 | 78.98 | 43.95 | 46.92 | 69.49 | 78.98
ImageNet (sup.) | 70.46 | 71.82 | 80.04 | 86.74 | 70.46 | 71.82 | 80.04 | 86.74
MoCo-v2 | ResNet-50 | 71.85 | 73.27 | 79.23 | 85.79 | 73.71 | 75.65 | 80.08 | 86.05
MoCo-v2+TP | 72.61 | 73.91 | 79.04 | 85.35 | 74.50 | 76.32 | 80.20 | 86.11
SeCo (ours) | 77.49 | 79.13 | 81.72 | 87.12 | 78.56 | 80.35 | 82.62 | 87.81

Table 1: Mean average precision on the BigEarthNet land-cover classification
task. Results cover different pre-training approaches and different ResNet
backbones. We also explore the effect of the unlabeled pre-training set size
between 100k and 1M images, and the size of the BigEarthNet training set
between 10% and 100%.

##  4 Experiments

In this study, we evaluate the learned representations on three downstream
tasks: two land-cover classification tasks, where the representation should be
invariant to seasonal changes, and a change detection task, where the
representation should be variant to seasonal changes.

##### Pre-training Implementation Details

We adopt Momentum Contrast (MoCo-v2) [6] as the backbone for our method due to
its combination of state-of-the-art performance and memory efficiency. We
apply the same artificial augmentations as MoCo-v2, i.e. color jittering,
random grayscale, Gaussian blur, horizontal flipping, and random-resized
cropping. We use a ResNet [17] architecture as the feature extractor, and a
2-layer MLP head with a ReLU activation and 128-dimensional output for each
embedding sub-space. We also use separate queues [18] for each embedding sub-
space, containing 16,384 negative embeddings at a time. We pre-train the
network for 200 epochs with a batch size of 256. We use an SGD optimizer with
a momentum of 0.9 and a weight decay of 1e-4. We set an initial learning rate
of 0.03 and divide it by 10 at 60% and 80% of the epochs. A temperature
scaling $\tau$ of 0.07 is used in the contrastive loss. Although the collected
dataset contains up to 12 spectral bands, in this work we focus on the RGB
channels since it is a more general modality.

##### Methods

We compare our unsupervised learning approach against several baselines,
including random initialization, ImageNet supervised pre-training, and self-
supervised pre-training. For the latter, we provide results for MoCo-v2 pre-
training on our unsupervised dataset without exploiting the temporal
information. In this case, the length of the dataset depends on the total
number of images and not the number of geographical locations, so we divide
the number of pre-training epochs by the number of images per location. We
also provide results for MoCo-v2 pre-training on our dataset leveraging the
temporal information for generating positive pairs (MoCo-v2+TP), i.e. positive
image pairs come from the same location at different times, and MoCo-v2
artificial augmentations are then applied to the spatially aligned image pairs
(similar to Ayush et al. [1]). We evaluate all methods with linear probing
(freezing the encoder and training only the classifier) and fine-tuning
(updating the parameters of both the encoder and the classifier).

###  4.1 Land-Cover Classification on BigEarthNet

BigEarthNet [38] is a challenging large-scale multi-spectral dataset of
Sentinel-2 [10] images, captured with similar sensors to the ones in our
unsupervised dataset, i.e. 12 frequency channels (including RGB) are provided.
It consists of 125 Sentinel-2 tiles acquired between June 2017 and May 2018
over 10 European countries, which are divided into 590,326 non-overlapping
image patches, each covering an area of $1.2\times 1.2$ km with resolutions of
10 m, 20 m, and 60 m per pixel. We discard about 12% of the patches which are
fully covered by seasonal snow, clouds or cloud shadows. This is a multi-label
dataset where each image is annotated by multiple land-cover classes, so we
measure the downstream performance in terms of mean average precision (mAP).
We adopt the new class nomenclature introduced in [39], and we use the same
train/val splits proposed in [30].

##### Implementation Details

We evaluate the learned representations by training a linear classification
layer with supervised learning. We initialize the ResNet backbone with a pre-
trained representation and add a single fully-connected layer which maps from
the intermediate representation to class logits. We fine-tune the network for
100 epochs with a batch size of 1024, and report the best validation results
for each run. We use an Adam optimizer with default hyper-parameters. For
linear probing, we set the initial learning rate to 1e-3; for full fine-
tuning, we set the initial learning rate to 1e-5. During training, the
learning rate is divided by 10 at 60% and 80% of the epochs.

##### Quantitative Results

Table 1 compares the accuracy of SeCo pre-training on BigEarthNet with other
pre-training methods. The comparison is done by linear probing or fine-tuning
with different backbones, number of pre-training images, and percentage of
BigEarthNet labeled data available. For linear probing, we observe that SeCo
consistently outperforms MoCo-v2+TP. We also observe that temporal positives
(TP) improve the performance of MoCo-v2 by a narrow margin. Moreover, we find
that SeCo features significantly improve over ImageNet pre-trained features,
which confirms our hypothesis that there is a gap between remote sensing and
natural image domains. We also find that this gap decreases when fine-tuning
an ImageNet pre-trained feature extractor on the whole BigEarthNet training
set. Nonetheless, with 1M images and a ResNet-50 backbone, SeCo features
achieve $1.1\%$ higher accuracy than ImageNet features. To the best of our
knowledge, this is the first time an unsupervised method obtains higher
accuracy than ImageNet pre-training on BigEarthNet with 100% of the labels.
Regarding the backbone size, we observe a wider performance gap between
ResNet-18 and ResNet-50 when linear probing than when fine-tuning the whole
network. We also find that pre-training with 1M images yields better
performance regardless of the backbone used. In all cases, we find that SeCo
is more efficient than the baselines when only using $10\%$ of BigEarthNet’s
labeled data; we provide more details in the next section.

Figure 3: Label-efficient land-cover classification on BigEarthNet. We use a
ResNet-18 backbone pre-trained on 1M images.

##### Study on Label-Efficient Transfer Learning

Figure 3 shows the linear probing and fine-tuning performance of SeCo and the
different baselines for different percentages of labeled data on BigEarthNet.
For linear probing, we observe that, with only $1\%$ of the BigEarthNet
labels, SeCo outperforms ImageNet pre-training with $100\%$ of the labels and
matches MoCo-v2 with $20\%$ of the labels. We also observe that the gap
between ImageNet pre-training and self-supervised learning increases with the
amount of labeled data, while the gap between self-supervised methods does not
change significantly. For all percentages of labeled data, SeCo achieves a
constant $\sim 4\%$ improvement gap with respect to MoCo-v2+TP. When fine-
tuning, the performance gap between self-supervised methods and ImageNet
narrows down when increasing the percentage of labeled data. Nevertheless,
SeCo is more label-efficient than all the baselines, matching the performance
of ImageNet pre-training using all the available labels with only $50\%$ of
the labels.

##### Ablation on the Locations Sampling Strategy

In order to evaluate the effectiveness of our sampling strategy to collect
uncurated images for pre-training remote sensing representations, we download
an alternative version of the SeCo dataset where the Earth locations are
sampled uniformly from within the continents. We download 100k images
following this approach and pre-train a ResNet-18 with the SeCo method. Table
2 compares transfer learning performance on the BigEarthNet downstream task
when using each sampling scheme. We observe that a SeCo representation pre-
trained on images sampled from a mixture of Gaussians centered around human
settlements (see section 3.1) provides better downstream performance than
sampling the images uniformly. We argue this is because populated regions tend
to be more diverse due to human activity and thus collected images contain
more information for learning good representations.

Sampling | Linear probing | Fine-tuning
---|---|---
10% | 100% | 10% | 100%
Gaussian | 74.67 | 75.52 | 81.49 | 87.04
Uniform | 71.63 | 72.59 | 79.65 | 85.75
Table 2: Comparison of SeCo dataset sampling strategies. We use a ResNet-18
backbone pre-trained on 100k images.

###  4.2 Land-Cover Classification on EuroSAT

EuroSAT [19] also addresses the challenge of land use and land cover
classification using Sentinel-2 satellite images. The images correspond to 34
European countries, and they consist of 10 classes corresponding to different
land uses. Each of the classes is composed of 2,000 to 3,000 images, making a
total of 27,000 labeled images. The images size is $64\times 64$ pixels,
covering an area of $640\times 640$ m. All 13 Sentinel-2 spectral bands are
included. We adopt the same train/val splits proposed in [30].

##### Implementation Details

On this task, we also evaluate the learned representations by learning a
linear classifier with supervised learning. We initialize a ResNet-18 backbone
with a pre-trained representation and add a single fully-connected layer on
top. In this case, we initialize the backbone with representations pre-trained
on 1M satellite images (except when using random weights or loading an
ImageNet pre-trained model). We freeze the backbone weights and train the
classifier for 100 epochs with a batch size of 32, reporting the best
validation accuracy for each run. We use an Adam optimizer with default
hyperparameters, setting the initial learning rate to 1e-3 and dividing it by
10 at 60% and 80% of the epochs.

##### Quantitative Results

Table 3 compares the linear probing accuracy of SeCo representations against
the different baselines. We see that SeCo achieves $6.7\%$ higher accuracy
than ImageNet pre-training and $3.6\%$ higher accuracy than MoCo-v2+TP. These
results confirm that the learned representation is not only effective on
BigEarthNet, but also generalizes to other remote sensing datasets such as
EuroSAT.

Pre-training | Accuracy
---|---
Random init. | 63.21
Imagenet (sup.) | 86.44
MoCo-v2 | 83.72
MoCo-v2+TP | 89.51
SeCo (ours) | 93.14
Table 3: Fine-tuning accuracy on the EuroSAT land-cover classification task.
We use a ResNet-18 backbone pre-trained on 1M images.

###  4.3 Change Detection on Onera Satellite

The Onera Satellite Change Detection (OSCD) dataset [8] is composed of 24
pairs of multispectral images from Sentinel-2. The images were recorded
between 2015 and 2018 from locations all over the world with various levels of
urbanization, where urban changes were visible. Each location contains aligned
pairs covering all 13 Sentinel-2 spectral bands. Images vary in spatial
resolution between 10 m, 20 m and 60 m, with approximately $600\times 600$
pixels at 10 m resolution. The goal is to detect changes between satellite
images from different dates. Pixel-level change ground truth is provided for
all training and validation image pairs. We use the same train/val splits
proposed by Daudt et al. [8]: 14 images for training and 10 images for
validation. We measure the the downstream performance in terms of F1 score, as
it is common in the image segmentation literature.

##### Implementation Details

For every pair of images from a given location at two different timestamps, we
produce segmentation masks by following a procedure similar to Daudt et al.
[7]. First, a ResNet-18 backbone extracts features from each image. We keep
the features after each downsampling operation in the backbone network. Then,
we compute the absolute value of the difference between the two sets of
features in each pair, and use the feature differences as input to a U-Net
[34] in order to generate binary segmentation masks. The backbone network is
initialized with representations pre-trained on 1M satellite images. To avoid
overfitting, we freeze the backbone and only train the weights of the U-Net,
add a $0.3$ dropout rate after each upsampling layer in the U-Net, and augment
the training images with random horizontal flips and $90^{\circ}$ rotations.
In addition, since the images in the OSCD dataset have variable size, we split
them into non-overlapping patches of $96\times 96$ pixels. We train the
decoder for 100 epochs with a batch size of 32, and report results on the
validation set from the point of view of the ”change” class. We use an Adam
optimizer with a weight decay of 1e-4. We set the initial learning rate to
1e-3 and decrease it exponentially with a multiplicative factor of 0.95 at
each epoch.

##### Quantitative Results

Table 4 compares SeCo with random initialization, ImageNet pre-training,
MoCO-v2, and MoCo-v2+TP. We observe that SeCo initialization achieves higher
recall and F1 score than all the baselines. In particular, SeCo outperforms
MoCo-v2+TP by 6.8% F1 score. This might be due to MoCo-v2+TP representations
being invariant to temporal variations, which is not a desirable property in a
change detection task. Interestingly, although both SeCo and MoCo-v2 consider
image patches from the same location at different timestamps as negative pairs
(i.e. their learned representations are variant to time), SeCo attains a 6.2%
higher F1 score. This indicates that the multiple embedding sub-spaces make
SeCo more effective at detecting temporal changes by disentangling image
augmentations from temporal variations.

Pre-training | Precision | Recall | F1
---|---|---|---
Random init. | 70.53 | 19.17 | 29.44
Imagenet (sup.) | 70.42 | 25.12 | 36.20
MoCo-v2 | 64.49 | 30.94 | 40.71
MoCo-v2+TP | 69.14 | 29.66 | 40.12
SeCo (ours) | 65.47 | 38.06 | 46.94
Table 4: Fine-tuning results on the Onera Satellite change detection task. We
use a ResNet-18 pre-trained on 1M images.

##### Qualitative Results

Figure 4 compares the change detection masks produced by our method and all
the baselines on two samples from the OSCD validation set. We observe that
SeCo pre-training produces higher quality masks which cover more of the
changed pixels without excessive false negatives. We also notice some
discrepancies in the performance of MoCo-v2 with and without leveraging
temporal information (TP). We hypothesize these might be due to the different
treatment of temporal invariance by each approach, and the image differences
resembling more artificial augmentations or temporal changes. SeCo overcomes
this problem by learning a representation that preserves time-varying and
invariant factors.

Figure 4: Comparison of qualitative results on the Onera Satellite change
detection task. Each row contains the input images, the ground truth mask, and
the generated change detection masks for a validation sample.

##  5 Related Work

##### Learning from Uncurated Data

Recent efforts in unsupervised feature learning have focused on either small
or highly curated datasets like ImageNet, whereas using uncurated raw datasets
was found to decrease the feature quality when evaluated on a transfer task
[9, 2]. Caron et al. [3] propose a self-supervised approach which leverages
clustering to improve the performance of unsupervised methods trained on
uncurated data. Other methods use metadata such as hashtags [23, 40, 27],
geolocation [45] or the video structure [14] as a source of noisy supervision.
In our work, we leverage the geographical and temporal information of remote
sensing data to learn unsupervised representations from uncurated datasets.

##### Multi-augmentation Contrastive Learning

Recent self-supervised contrastive learning methods have been able to produce
impressive transferable visual representations by learning to be invariant to
different image augmentations. However, these methods implicitly assume a
particular set of representational invariances, and can perform poorly when a
downstream task violates this assumption. Xiao et al. [47] propose Leave-one-
out Contrastive Learning (LooC), a multi-augmentation contrastive learning
framework that produces visual representations able to capture varying and
invariant factors by constructing separate embedding spaces, each of which is
invariant to all but one augmentation. In our work, we use a similar approach
to learn representations that are variant and invariant to the seasonal
changes present in remote sensing images.

##### Unsupervised Learning in Remote Sensing

While unsupervised learning has been extensively studied on natural image
datasets (e.g. ImageNet), this subfield remains underexplored on the remote
sensing domain. This is quite surprising given the importance of remote
sensing for Earth observation, the vast amount of readily available data, and
the many opportunities for self-supervision from the unique characteristics of
satellite images. For instance, Jean et al. [22] use the geographical
information of images to sample positive and negative pairs and build a
pretext task based on the triplet loss. Uzkent et al. [42] pair georeferenced
Wikipedia articles with satellite images of the corresponding locations, and
learn representations by predicting properties of the articles from the
images. Vincenzi et al. [43] leverage the multi-spectrality of remote sensing
images to build a colorization task, where they reconstruct the visible colors
from the other spectral bands. More similar to our work, Ayush et al. [1] also
propose to exploit the temporal information in satellite imagery to generate
positive pairs and train a contrastive objective. However, their
representations are always invariant to temporal changes, which might be
detrimental for downstream tasks involving temporal variation. We overcome
this problem by using multi-augmentation contrastive learning, where the
representations preserve time-varying and invariant information.

##  6 Conclusions

We presented Seasonal Contrast (SeCo), a new transfer learning pipeline for
remote sensing imagery. SeCo consists of a data collection strategy and a
self-supervised learning algorithm that leverages this data. First, we sample
locations around populated regions over multiple timestamps, which provides a
diverse set of satellite images. Then, we extend multi-augmentation
contrastive learning methods to take into account the seasonal changes and
learn rich and transferable remote sensing representations.

We compared SeCo with the common ImageNet pre-training and MoCo pre-training
on the collected data using different backbones and dataset sizes. We found
that SeCo outperforms the considered baselines on BigEarthNet, EuroSAT and
OSCD tasks. Thus, we conclude that domain-specific unsupervised pre-training
is more effective for remote sensing applications than pre-training with
standard datasets such as ImageNet or algorithms such as MoCo.

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# Cohomology in nonunitary representations of semisimple Lie groups (the group
$U(2,2)$)

A. M. Vershik St. Petersburg Department of Steklov Institute of Mathematics
and St. Petersburg State University, St. Petersburg, Russia. Email:
<EMAIL_ADDRESS>Supported by the RFBR grants 11-01-00677-a and
13-01-12422-ofi-m. M. I. Graev Institute for System Studies, Moscow, Russia.
Email<EMAIL_ADDRESS>Supported by the RFBR grant 13-01-00190 .

###### Abstract

We suggest a method of constructing special nonunitary representations of
semisimple Lie groups using representations of Iwasawa subgroups. As a typical
example, we study the group $U(2,2)$.

To the 100th birthday of our teacher Israel Moiseevich Gelfand

##  1 Introduction: a survey of the theory of special representations

###  1.1 Groups of currents and special representations

The group of currents $G^{X}$, where $X$ is a topological space equipped with
a Borel probability measure $m$ and $G$ is an arbitrary locally compact group,
is the group of continuous maps $X\rightarrow G$ with pointwise multiplication
and with some integrability condition with respect to $m$. The study of
representations of such groups is inspired both by representation theory
itself and applications to mathematical physics. A well-known model of
irreducible representations of current groups $G^{X}$ is the Fock, or
Gaussian, model, in which a crucial role is played by nonzero first cohomology
of the group of coefficients $(G$) with values in irreducible unitary
representations of this group. Such an irreducible representation $T$ of the
group $G$ in a Hilbert space $H$ has a nontrivial 1-cocycle, i.e., a
continuous map $b:G\rightarrow H$ satisfying the condition

$b(g_{1}g_{2})=T(g_{1})b(g_{2})+b(g_{1})\quad\mbox{for any}\quad
g_{1},g_{2}\in G$

and the following nontriviality condition: there is no vector $\xi\in H$ such
that $b(g)=T(g)\xi-\xi$ for every $g\in G$. Such representations are called
special. The trivial representation is special for all groups having
nontrivial complex characters, since an additive complex character is exactly
a nontrivial 1-cocycle with values in the one-dimensional trivial complex
representation.

In other terms, a special representation is a representation in which there
exist almost invariant vectors; the latter term means, for a representation of
a group $G$ in a space $H$, that for every $\epsilon>0$ and every compact
subset $K$ in $G$, there exists a vector $h$ in $H$ such that
$\|U_{k}h-h\|<\epsilon$ for every $k\in K$, where $U_{k}$ is the operator in
the representation corresponding to the element $k$.

By a theorem from [25], every special representation of a compactly generated
group is not Hausdorff separated from the trivial representation in the Fell
topology. The Fell topology on the space of unitary representations of a
locally compact group $G$ is defined as follows: an open neighborhood of a
representation $\pi$ in a Hilbert space $H$ is determined by a number
$\epsilon$, a finite collection $h_{1},\dots h_{k}$ of elements of $H$, and a
compact subset $K$ in $G$; it consists of all unitary representations $\rho$
of $G$ such that the space of $\rho$ contains elements $f_{1},\dots,f_{k}$
such that

$\sup_{i=1,\dots,k;\;g\in
K}\left\\{|\langle\pi(g)h_{i},h_{i}\rangle-\langle\rho(g)f_{i},f_{i}\rangle|<\epsilon\right\\}.$

The converse is not true: a representation that is not separated from the
trivial representation is not necessarily special, i.e., does not necessarily
have non vanishing first cohomology.

In [13], Y. Shalom proved the conjecture stated in [25]: if for a locally
compact group the trivial representation is not isolated in the space of all
irreducible unitary representations (i.e., the group does not have Kazhdan’s
property (T)), then it has at least one special representation. The proof of
the main theorem in [13] (see also [2, Theorem 3.2.1]) is not constructive,
and hence does not provide a direct method of finding a special
representation: we know very little about how to select special
representations from the set of all representations that are not separated
from (or “glued” to) the trivial representation.

We use the following terminology. The core of a given representation $\pi$ is
the set of all representations $\rho$ such that arbitrary open neighborhoods
of $\pi$ and $\rho$ have a nonempty intersection. The core of the regular
representation of an amenable group contains all irreducible representations,
and this is a characteristic property of amenability. This notion is of
special importance for irreducible representations and, in particular, for the
trivial representation. Unless otherwise stated, by the core of a group we
mean the core of its one-dimensional trivial representation.111A more detailed
terminology is as follows. Representations lying in the core of the trivial
representation are called infinitesimal (in a Leibniz-like sense, see [25]);
the subcore of a given representation $\pi$ is the set of all representations
$\rho$ whose closure contains $\pi$ (correspondingly, the subcore of a group
is the subcore of its trivial representation). In other terms, this means that
$\rho$ weakly contains $\pi$. The subcore is, obviously, a subset of the core.
The interpretation of these notions in terms of Hausdorff’s separation axioms
is as follows: if the trivial representation lies in the core of $\rho$, then
$T_{2}$ does not hold for these two elements; if $\rho$ lies in the subcore,
even $T_{1}$ does not hold; $T_{0}$ always holds, since the trivial
representation is closed.

One says that nonvanishing cohomology with values in a representation is
reduced (see [13]) if the corresponding cocycle is not a limit of trivial
cocycles. This can be the case only for representations lying in the core, but
not in the subcore. For the semisimple groups $O(n,1),U(n,1)$, the non
vanishing cohomology is reduced. The authors do not know whether there exist
solvable groups satisfying this condition. Finally, for completeness we
mention another interesting notion — that of groups with the Haagerup property
(see [2, 8]): these are groups for which the trivial representation is not
isolated (i.e., which do not have Kazhdan’s property) but cocycles with values
in a special representation satisfy a certain nondegeneracy condition:
regarded as a map

$\beta:G\rightarrow H$

from the group to a Hilbert space, the cocycle is proper, i.e., the preimages
of bounded sets in $H$ are precompact in the group. This property holds for
all amenable groups, free groups ([8]), the groups $O(n,1),U(n,1)$, and others
(see [4]). Examples of groups that satisfy neither Kazhdan’s property nor the
Haagerup property are not yet sufficiently studied.

###  1.2 Special representations of rank $1$ groups and their solvable
subgroups

In the most important class of Lie groups — that of semisimple groups — only
the groups $O(n,1)$ and $U(n,1)$, which are of rank $1$, have special
representations. The other semisimple groups (including, as proved by Kostant
[10], even rank $1$ groups $Sp(n,1)$) have Kazhdan’s property (T): their
trivial representations are isolated, and the Fock model of constructing
representations of the corresponding groups of currents does not apply. Note
that Kostant’s theorem on $Sp(n,1)$ still has no geometric proof.

One may say that the analysis of representations of the groups of currents for
$O(n,1)$ and $U(n,1)$ is developed quite well. Studies in this direction began
from the pioneer work by I. M. Gelfand and the authors of this paper, see [17,
18, 6]. The general scheme of the Fock model, regardless the concrete group,
was earlier considered by Araki [1] (see [12]), however, before the paper [17]
there were no examples of semisimple groups for which the core is nontrivial.
In these papers, as well as in [9, 5, 7, 3], irreducibility conditions for
representations of current groups were found, and other properties of these
groups were established. The key role was played by the ideas of the paper
[6], in which a method was suggested, for semisimple groups of rank $1$, of
reducing the problem to a solvable subgroup, on the example of $SL(2,R)$. The
elaboration of this idea by the authors of this paper during the last 10 years
has led to new models of representations of current groups which are
equivalent to the Fock one but are constructed from other (non-Gaussian) Lévy
measures (see [19, 20]). This has led to constructing the integral, and then
Poisson and quasi-Poisson, models of representations of current groups ([21,
22]). There has also appeared the so-called infinite-dimensional Lebesgue
measure [14, 15, 16], which is the most precise continual analog of the
Lebesgue measure on a finite-dimensional positive octant. This measure is
closely related to the gamma process and is of great interest in itself.

###  1.3 Solvable subgroups of semisimple groups, and a refinement of the
Iwasawa decomposition

It is well known that for commutative and nilpotent groups the special
representations are exhausted by the trivial representation (see [7, 5]). But
even for solvable groups, this problem is not sufficiently studied. We are
interested in a concrete class of solvable groups, which are subgroups of
$O(p,q),U(p,q)$, and our first example is the group $U(2,2)$. We are working
with a solvable subgroup of $U(2,2)$ which should be called the Iwasawa group.
Such a subgroup can be defined in an arbitrary connected semisimple real Lie
group (see [11], where it was called the “maximal connected triangular
subgroup”). The Iwasawa decomposition means that this subgroup is
“complementary” to the maximal compact subgroup.

A solvable subgroup (like any amenable group) has a special representation, so
that we obtain the following strategy of constructing a representation of the
group of currents of a semisimple group $G$: first find a special
representation, unitary or not, of this solvable subgroup and construct a
representation of the group of its currents, and then try to extend the
special representation to the whole semisimple group $G$ and construct an
extension of the representation to the group of currents of $G$. As mentioned
above, for groups of rank $1$, this trick was first used in [6] in the case of
$SL(2,R)$, and then studied in detail in a recent series of papers of the
authors ([21, 22]). When passing from groups of rank $1$ to higher ranks, this
idea is still working.

Here we consider this problem on the concrete example of the group $U(2,2)$,
keeping in mind the more general situation, which will be considered
elsewhere. We describe in detail the Iwasawa subgroup (denoted by $P$ in what
follows) for this case. The first question is, what are its special
representations? But here we encounter a new problem: for our plan to be
viable, this special representation of the Iwasawa subgroup must be
faithful.222A faithful (or nondegenerate) representation is a representation
whose kernel is trivial. The authors do not know for what groups this is the
case. For example, nilpotent groups have no faithful special representations.
This fact is of interest already for the Heisenberg group, and it is
equivalent to a version of the uncertainty principle.333From discussions with
V. P. Khavin, the first author inferred that this fact about the Heisenberg
group apparently still has no purely analytical (rather than representation
theoretic) proof. For the affine group $\mbox{Aff}(\mathbb{R})$, the infinite-
dimensional representation (which is quasi-equivalent to the regular one) is a
faithful special representation. But even for simplest three-dimensional
solvable groups (in particular, for the group $S$, see below), the question is
not trivial.

Problem. What groups (in particular, what solvable groups) have a faithful
irreducible unitary special representation? More precisely: when does such a
representation exist for the Iwasawa subgroup of a semisimple real Lie group?

Thus the main difficulty is to construct a faithful special representation,
unitary or not, of the Iwasawa subgroup. A construction of a nonunitary
faithful special representation of the group $U(2,2)$ is the main result of
this paper, and the authors have no doubts that such a construction can be
carried out for every real semisimple group. This makes it possible to extend
the cocycle to the whole semisimple group and construct a representation of
the group of currents, since, as follows from the results of the above-
mentioned papers, there are models (for instance, the Poisson model, unlike
the Fock one) of representations of groups of currents that do not in any way
rely on the unitarity of the original representation. The question of whether
a cocycle can be constructed with values in a unitary representation of the
Iwasawa subgroup is yet open.

In conclusion of this survey, we mention a somewhat different approach, which
is closer to the original work on groups of currents and classical work on the
representation theory of semisimple groups. Namely, it is well known that
special representations of semisimple groups of rank $1$ lie in the “tail” of
the complementary series, and complementary series exist for every semisimple
group. However, for groups of rank greater than 1, unitary representations do
not “reach” the trivial representation; more exactly, unitarity gets lost
under deformations of the trivial representation. On the other hand, it is
known that in some cases one can find an indefinite bilinear form invariant
under the action of the group in such a nonunitary representation. The
properties of the corresponding space with an indefinite metric are poorly
studied, and the problem of the existence of a special representation, perhaps
nonunitary, apparently has not been stated. Our alternative strategy relies on
the analytic continuation of unitary representations of the Iwasawa subgroups
and, in particular, construction of a nonunitary special representation within
this framework. It is conceivable that these two approaches to the
representation theory of the groups of currents for semisimple groups of rank
greater than 1 may lead to different classes of representations.

##  2 The group $U(2,2)$ and its Iwasawa subgroup

###  2.1 Description of the group $U(2,2)$

As a first example, we consider the group $U(2,2)$ of linear transformations
of ${\mathbb{C}}^{4}$ preserving a fixed Hermitian form with signature
$(2,2)$; here we choose the Hermitian form

$x_{1}\bar{x}_{3}+\bar{x}_{1}x_{3}+x_{2}\bar{x}_{4}+\bar{x}_{2}x_{4}.$

The group $U(2,2)$ is one of the simplest examples of a semisimple Lie group
whose real rank is greater than one (it is equal to 2). The groups of the form
$U(p,q)$ are called pseudo-unitary. In this section, we refine the Iwasawa
decomposition for this group and study the structure of the key object, the
solvable Iwasawa subgroup $P$.

We will write elements of the group $U(2,2)$ as $2\times 2$ block matrices
with $2\times 2$ blocks:

$\left(\begin{array}[]{cc}g_{11}&g_{12}\\\ g_{21}&g_{22}\\\
\end{array}\right),$

where $g_{ij}$ are complex $2\times 2$ matrices satisfying the relation

$g\sigma
g^{*}=\sigma,\qquad\mbox{with}\quad\sigma=\left(\begin{array}[]{cc}0&e_{2}\\\
e_{2}&0\\\ \end{array}\right).$

Here $e_{2}$ is the $2\times 2$ identity matrix and $*$ stands for the
conjugate transpose in the complex case, and for the transpose in the real
case.

These relations are equivalent to the following relations between the blocks
of the matrix $g$:

$\displaystyle g_{12}g^{*}_{21}+g_{11}g^{*}_{22}$ $\displaystyle=$
$\displaystyle e_{2},$ $\displaystyle g_{11}g^{*}_{12}+g_{12}g^{*}_{11}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
g_{22}g^{*}_{21}+g_{21}g^{*}_{22}$ $\displaystyle=$ $\displaystyle 0.$

The real dimension of the group $U(2,2)$ is equal to 16. In what follows, the
key role is played by the solvable subgroup $P$ of $U(2,2)$ generated by the
following two subgroups $N$ and $S$:

  * •

the additive (commutative) group $N$ of skew-Hermitian block matrices of the
form

$\left(\begin{array}[]{cc}e_{2}&0\\\ n&e_{2}\\\ \end{array}\right),$

where $n$ is a skew-Hermitian $2\times 2$ matrix: $n+n^{*}=0$;

  * •

the solvable subgroup $S$ (of derived length 2) of block matrices of the form

$\left(\begin{array}[]{cc}s^{*{-1}}&0\\\ 0&s\\\ \end{array}\right),$

where $s$ is a lower triangular complex matrix with positive diagonal entries:

$\left(\begin{array}[]{cc}r_{1}&0\\\ r&r_{2}\\\ \end{array}\right),\qquad
r_{1},r_{2}>0,\;r\in\mathbb{C}.$

The real dimension of the groups $S$ and $N$ is equal to 4, and that of the
group $P$ is equal to 8.

A general element of the group $P$ (a pair $(s,X)$) is as follows:

$\left(\begin{array}[]{cc}s^{*{-1}}&0\\\ X&s\\\ \end{array}\right),$

where $X$ is a $2\times 2$ matrix satisfying a condition that could be called
the relative skew-Hermiticity (with respect to the matrix $s$):

$sX^{*}+Xs^{*}=0.$

The following assertion can be checked directly.

######  Proposition 1.

The group $U(2,2)$ is algebraically generated by the elements of the group $P$
and the involution $\sigma$; the intersection of the groups $N$ and $S$
consists of the identity element.

The homogeneous space $U(2,2)/K$ where $K$ is the maximal compact subgroup of
$U(2,2)$ is exactly the space of the subgroup $P$, which justifies calling it
the Iwasawa subgroup. (This makes it possible to extend a cocycle from $P$ to
the whole group $U(2,2)$, see below.)

Since the group $S$ acts in an obvious way on the additive group $N$ of skew-
Hermitian matrices according to the rule

$n\mapsto sns^{*},\qquad s\in S,\;n\in N,$

we can define the semidirect product $Q=S\rightthreetimes N$ of $S$ and $N$.
One can directly check the following important assertion.

######  Theorem 1.

The groups $P$ and $Q$ are canonically isomorphic. The isomorphism
$I:P\rightarrow Q$ is given by the formula

$I:(s,X)\rightarrow(s,Xs^{*})$

(the left-hand side is an element of the subgroup $P\subset U(2,2)$, and the
right-hand side is an element of the semidirect product $Q$, the matrix
$Xs^{*}$ being obviously skew-Hermitian). The inverse isomorphism is as
follows: $(s,n)\rightarrow(s,{ns^{*}}^{-1}),$ where $n$ is skew-Hermitian.

###### Proof.

We check that the multiplication is homomorphic:

$I((s_{1},X_{1})\circ(s_{2},X_{2}))=I(s_{1}s_{2},X_{1}{s_{2}^{*}}^{-1}+s_{1}X_{2})=$
$=(s_{1}s_{2},X_{1}s_{1}^{*}+s_{1}X_{2}s_{2}^{*}s_{1}^{*})=I(s_{1},X_{1})I(s_{2},X_{2}).$

∎

The map $I$ sends $P$ to the group whose matrix realization consists of the
collection of pairs $(s,X)$ satisfying the above condition. Hence the
isomorphism of the semidirect product $Q$ of $S$ and $N$ with its matrix
realization, i.e., the group $P$, does not coincide with the direct product of
the identity isomorphism of the subgroup $S$ and the identity isomorphism of
the normal subgroup $N$. Nevertheless, the group $P$ is the semidirect product
$S\rightthreetimes N$.444In the case of a group of rank $1$, the matrix
realization of the Iwasawa subgroup is somewhat simpler; for example, for the
group $SL(2)$ the Iwasawa subgroup $P$ is the subgroup of triangular matrices
$\left(\begin{array}[]{cc}s^{-1}&o\\\ n&s\\\ \end{array}\right),\qquad
s\in{\mathbb{R}}_{+},\;n\in\mathbb{R},$ and the structure of the semidirect
product agrees with the ordinary matrix representation, since the group
regarded as a set is the direct product of the normal subgroup
$\left(\begin{array}[]{cc}1&0\\\ n&1\\\ \end{array}\right)$ and the subgroup
$\left(\begin{array}[]{cc}s^{-1}&0\\\ 0&s\\\ \end{array}\right).$

A general element of the commutant of the group $S$ in the above notation is
as follows:

$\left(\begin{array}[]{cc}1&0\\\ r&1\\\ \end{array}\right),$

where $r$ is a complex number. The derived length of the solvable group $S$ is
equal to 2. Thus the commutant of the group $P$ is the semidirect product of
$\mathbb{C}$ and the commutative group of skew-Hermitian matrices (with a
nontrivial action of $\mathbb{C}$).

######  Corollary 1.

The derived length of the solvable group $P$ is equal to $3$.

###  2.2 Representations of the Iwasawa subgroup and almost invariant measures

Our aim is to study the special representation of the subgroup $P$ and then
extend it to the whole group $U(2,2)$. We will study representations of the
group $P$ regarded as a semidirect product.

The group $N$ is isomorphic to the group $\hat{N}$ of all its continuous
characters, and we have the conjugate action of the group $S$ on $\hat{N}$;
moreover, the direct and conjugate actions on the group $N$ (identified with
its dual group) coincide:

$n\rightarrow
sns^{*};\qquad\chi\rightarrow\chi_{s}\quad\mbox{where}\quad\chi_{s}(n)=\chi(sns^{*}).$

######  Proposition 2.

The action of the group $S$ by automorphisms on the group $\hat{N}$ (and on
the group $N$) has four orbits of positive measure; they are parametrized by
the signs of the imaginary parts of the diagonal entries, i.e., are the orbits
of the elements

$\left(\begin{array}[]{cc}\pm i&0\\\ 0&\pm i\\\ \end{array}\right),$

respectively. On every orbit, the action of $S$ is free and faithful, so that
each orbit can be identified with the group $S$; then the action coincides
with the action of $S$ on itself by right translations.

Note that there are also orbits of smaller dimension, which have zero measure,
but we will not need them.

It is clear that all four actions of the group $S$ on the nondegenerate orbits
are topologically isomorphic and the corresponding representations of the
semidirect product differ only by a ${\mathbb{Z}}_{2}$-valued cocycle which
acts as a multiplication, so that it suffices to consider only one (any)
orbit.

A unitary representation of the semidirect product of a group and a
commutative group (in our case, $P=S\rightthreetimes N$) has the following
canonical realization. Consider a probability measure $\mu$ on the group of
characters (i.e., on $\hat{N}$) that is quasi-invariant with respect to the
action (of the group $S$). All such measures are equivalent, since $S$ is
locally compact and its action is transitive. Hence in the Hilbert space
$L^{2}_{\mu}(\hat{N})$ we can define the unitary representation of the group
$P$ induced by the above action of $S$ and the representation of $N$ in which
an element of $N$ acts as the multiplication by the corresponding character or
$\hat{N}$. General representations are realized in the vector-valued space
$L^{2}_{\mu}(\hat{N})$, but we do not consider them.

The irreducibility of the above representation of the semidirect product is
equivalent to the ergodicity of the measure $\mu$, and, by the above, we could
assume that the orbit is the group $S$ itself, i.e., consider the
representation of $S$ in the space $L^{2}_{\mu}(S)$ over a measure $\mu$
quasi-invariant with respect to the right action of the group.

These representations belong to the core of the group $P$; this follows from
the fact that each of them is quasi-equivalent to the regular representation
of the group, which (by the amenability of $S$) weakly contains the trivial
representation. Besides, the core contains the trivial representation, as well
as the special representations of the group $S$, which can be regarded as
representations of $P$, since $S=P/N$. It is not known whether the core is
exhausted by these representations, nor whether the four constructed
representations are special for the group $P$, or, in other words, whether
they have an almost invariant vector. But first we find out the structure of
the special representations of the group $S$.

######  Lemma 1.

The group $S$ has a continuum of unitary representations parametrized by the
points of $\mathbb{C}$ lying on the unit circle (characters). It has no
faithful special representations.

###### Proof.

The affine group $\mbox{Aff}(\mathbb{R})$, i.e., the group of matrices

$\left(\begin{array}[]{cc}e^{a}&0\\\ b&e^{-a}\\\ \end{array}\right),\qquad
a,b\in{\mathbb{R}},$

is the Iwasawa subgroup for $SL(2,R)$ and plays the same role as the group $P$
does for $U(2,2)$. Note that, regarded as a subgroup of $SL(2,R)$, it is the
semidirect product of ${\mathbb{R}}_{+}$ and $\mathbb{R}$ that agrees with the
matrix representation; as mentioned above, in our situation this is not the
case. It has (two) faithful special unitary representations, and they can be
extended to a special representation of $SL(2,R)$.

Consider the group $S$:

$\left(\begin{array}[]{cc}r_{1}&0\\\ r&r_{1}\\\ \end{array}\right),\qquad
r\in{\mathbb{C}},\quad r_{1},r_{2}>0;$

note that if we fix a value of the determinant ($=r_{1}\cdot r_{2}$) and a
unitary character on the group ${\mathbb{C}}={\mathbb{R}}^{2}$, the group $S$
can be mapped isomorphically to the group $\mbox{Aff}(\mathbb{R})$, and hence
all special representations of $\mbox{Aff}(\mathbb{R})$ can be lifted to
representations of $S$; all of them are not faithful; the group $S$ has no
other special representations. ∎

However, we are interested not as much in the group $S$, but in the group $P$.
Conjecturally, special unitary representations of $P$ can be constructed using
special (nonfaithful) representations of $S$ together with the representations
of $P$ constructed above. At the moment, the question of whether such
representations exist is open. For completeness, we describe a model of the
Hilbert space of a special unitary representation for the group $SL(2,R)$ and
its triangular subgroup ($P$). Consider the space
$L^{2}_{m}({\mathbb{R}}_{+})$ (where $m$ is the Lebesgue measure on the half-
line). It is more convenient to pass to the Fourier transform, and then the
required representation of the triangular subgroup (written as the group of
transformations $x\mapsto e^{\beta}x+a$ with $\beta,a\in\mathbb{R}$) can be
realized in $L^{2}_{\hat{m}}(\mathbb{R})$ (where ${\hat{m}}$ is the Lebesgue
measure on the line) as follows:

$(U_{a,\beta}F)(z)=\exp\\{iae^{z}\\}F(z+\beta),\quad z\in\mathbb{R}.$

An almost invariant vector in this model is an arbitrary function $f$
satisfying, for any $t,a,b\in\mathbb{R}$, the conditions

$f(x)=0\mbox{ if }x>t\in{\mathbb{R}};$ $f\notin
L^{2};\quad(1-\exp\\{ie^{z}b\\})f\in L^{2};\quad[f(\cdot)-f(\cdot+a)]\in
L^{2}.$

Another, more popular, description of the special representation (see [17]) in
a space of analytic functions is related to the limit of complementary series
representations as they tend to the trivial representation.

###  2.3 Almost invariant measures and nonunitary representations

We say that a measure $\nu$ on a group $S$ is (right) almost invariant if it
is infinite, absolutely continuous with respect to the right Haar measure on
$S$ (and hence quasi-invariant with respect to the right translations
$s\mapsto ss_{0}$), and its derivatives $\frac{d\nu(ss_{0})}{d\nu(s)}$ are
defined and bounded for every $s_{0}\in S$. (By the above isomorphism
$S\rightarrow H$, where $H$ is an arbitrary nondegenerate $S$-orbit on the
group of characters $\hat{N}$, this definition can be translated to measures
concentrated on any of the nondegenerate orbits of the group $\hat{N}$).

The almost invariance condition is obviously satisfied for the Lebesgue
measure on $S$, i.e.,

$ds=ds_{11}ds_{22}ds_{21}d\bar{s}_{21},$

since $d(ss_{0})=\pi(s)ds$, where $\pi(s)=s_{11}^{3}s_{22}$. It follows that
this condition holds for any measure of the form $d\nu(s)=u(s)ds$, where
$u(s)$ is an arbitrary function such that the ratio $\frac{u(ss_{0})}{u(s)}$
is a bounded function for every $s_{0}\in S$. In particular, it holds for the
measure $\mu$ that is invariant under the right translations (the Haar
measure):

$d\mu(s)=\pi^{-1}(s)ds,\quad\pi(s)=s^{3}_{11}s_{22}.$

However, for our purposes it is convenient to consider another measure.

Assume that a group $G$ acts on a space $X$, and we are given two equivalent
$G$-quasi-invariant measures $\mu$ and $\nu$ on $X$. Assume that the density
of one measure with respect to the other one is bounded away from zero and
infinity. In the spaces $L^{2}_{\mu}(X)$ and $L^{2}_{\nu}(X)$ we consider the
representation of the group $G$ by the substitutions
$(U_{g}f)(\cdot)=f(g\cdot)$ and the natural representation of the group of
multiplicators with absolute value equal to one. The well-known isometry
between these spaces, which multiplies a function by the square root of the
density of one measure with respect to the other one, commutes with the
multiplicators, but, in general, does not commute with the action of the
group. This isometry is widely used to correct the action; for instance, if
one of the measures is invariant, and thus determines a unitary representation
of the cross product, then the corrected action also becomes unitary.

Let $\nu$ be an almost invariant measure on $H$; consider a nonunitary
representation of the group $P$ in the Hilbert space $L^{2}(S,\nu)$, i.e., the
space of functions $F$ on $S$ with the norm
$\|F\|^{2}=\int_{S}|F(s)|^{2}d\nu(s)<\infty$. By definition, the operators
corresponding to the elements of the subgroups $N$ and $S$ are given by the
following formulas:

$\displaystyle(T(n)F)(s)$ $\displaystyle=$
$\displaystyle\chi_{k}(n,s)F(s)\quad\mbox{ for }n\in N;$ (1)
$\displaystyle(T(s_{0})F)(s)$ $\displaystyle=$ $\displaystyle
F(ss_{0})\quad\mbox{ for }s_{0}\in S.$ (2)

Here $\chi_{k}(n,s)$ is the image of a character $\chi(\cdot)\in\hat{N}$
regarded as a function on $S$ under the (unique) isomorphism between the orbit
of $S$ in $\hat{N}$ indexed by $k=1,2,3,4$ and $S$ preserving the action of
$S$; the difference between the four orbits reduces to multiplying the image
by $\pm i$ in each of the variables. It follows from the definition that the
operators $T(n)$ are unitary and the operators $T(s_{0})$ are bounded, by the
almost invariance of the measure $\nu$. One can rewrite the formulas in a more
compact form:

$\displaystyle(T(n)f)(s)$ $\displaystyle=$
$\displaystyle\chi(sns^{*})f(s)\quad\mbox{ for }n\in N;$ (3)
$\displaystyle(T(s_{0})f)(s)$ $\displaystyle=$ $\displaystyle
f(ss_{0})\quad\mbox{ for }s_{0}\in S,$ (4)

where $\chi$ is a fixed character of the $S$-orbit $H$ on the group of
characters $\hat{N}$.

It is not difficult to verify that the operators of the subgroups $N$ and $S$
together generate a representation of the whole group $P$ in the space
$L^{2}(S,\nu)$. In particular, if $\nu=\mu$ is the Haar measure on $S$, this
representation is unitary. Denote these representations of the group $P$ by
$\pi_{k}$, $k=1,2,3,4$; since all four representations essentially differ from
one another only by a choice of a character on the orbits, we omit the index
$k$.

######  Theorem 2.

The nonunitary representations $\pi$ of the group $P$ defined above are
operator irreducible and space irreducible. The representations corresponding
to different measures and different indices $k$ are space (unitarily)
equivalent if and only if the measures $\nu$ differ by a factor
$(\nu^{\prime}=c\nu)$ and the indices $k$ coincide.

An important question for us is how the cohomology depends on the measure. We
emphasize that changing the measure and, in particular, the unitarization of
representations (see above) does not induce an isomorphism of the cohomology
groups $H^{1}(G,\pi_{\mu})$ and $H^{1}(G,\pi_{\nu})$ where $\pi_{\mu}$,
$\pi_{\nu}$ are the representations corresponding to the measures $\mu$ and
$\nu$, since a space isometry does not in general send a cocycle of a group
with values in one space to a cocycle with values in another space. In other
words, for a given action of the group, the cohomologies with values in the
Hilbert space $L^{2}_{\mu}(X)$ for various almost invariant measures $\mu$ are
in general different. In the next section we choose an almost invariant
measure for which the cohomology is nontrivial.

###  2.4 A faithful nonunitary special irreducible representation of the
Iwasawa group

An important question for us is how the cohomology with values in
$L_{\nu}^{2}(X)$ depends on the almost invariant measure $\nu$. We emphasize
that changing the replacement of the measure by an equivalent one and, in
particular, the unitarization of representations (see above) does not induce
an isomorphism of the cohomology groups $H^{1}(G,\pi_{\mu})$ and
$H^{1}(G,\pi_{\nu})$, where $\pi_{\mu}$, and $\pi_{\nu}$ are the
representations corresponding to the measures $\mu$ and $\nu$, since a space
isometry does not in generally send a cocycle of a group with values in one
space to a cocycle with values in another the other space. In other words, for
a given action of the group, the cohomologies with values in the Hilbert space
$L^{2}_{\mu}(X)$ for various almost invariant measures $\mu$ are in generally
different. This explains our choice of an almost invariant measure in what
follows. In the next section we choose an almost invariant measure for which
the cohomology is nontrivial.

Let us fix an almost invariant measure $\nu$ on the space $S$ and introduce
the space $Z_{\nu}$ of measurable functions on $S$ satisfying the two
conditions

$\displaystyle\int_{S}|f(ss_{0})-f(s)|^{2}\,d\nu(s)$
$\displaystyle<\infty\quad\text{for any}\,s_{0}\in S,$
$\displaystyle\int_{S}|(\chi(sns^{*})-1)f(s)|^{2}\,d\nu(s)$
$\displaystyle<\infty\quad\text{for any}\,n\in N.$

Obviously, $L^{2}_{\nu}(S)\subset Z_{\nu}$.

We treat the elements of $f\in Z_{\nu}$ as coboundaries and the space of
functions of the form $b(g)=T(g)f-f$ as the space of cocycles. If $f\in
L^{2}_{\nu}(S)$, then $b(g)=T(g)f-f$ is a cocycle cohomologous to zero. This
implies the following assertion.

######  Lemma 2.

A representation of the group $P$ in the space $L^{2}_{\nu}(S)$ is special if
and only if

$L^{2}_{\nu}(S)\varsubsetneq Z_{\nu},$

i.e., if there exists a coboundary not lying in $L^{2}_{\nu}(S)$. Thus, the
first cohomology group has the form

$H^{1}(P;L^{2}_{\nu}(S)=Z_{\nu}/Z_{\nu}L^{2}_{\nu}(S).$

A measure $\nu$ is said to be special if $H^{1}(P;L^{2}_{\nu}(S))\neq 0$,
i.e., if there exist coboundaries not lying in $L^{2}_{\nu}(S)$. The authors
do not know whether the Haar measure $m$ on $S$ is special and, thereby,
whether the natural unitary representation of the group $P$ on $L^{2}_{m}(S)$
is special. For this reason, we use a different almost invariant measure to
construct a special, but not unitary, representation of $P$.

In what follows, we fix an almost invariant measure $\nu$ of the following
form:

$d\nu(s)=|s|^{-4}\,ds,\quad\text{where
}\,|s|^{2}=tr(s^{*}s)=s_{11}^{2}+s_{22}^{2}+|s_{21}|^{2}.$

It is convenient to write this measure in polar coordinates on $S$. To do this
end, note that the variety of elements $\omega\in S$ with norm $|\omega|=1$ is
equivalent to a domain on the unit sphere in $\mathbb{R}^{4}$. We define
spherical coordinates of a matrix $s\in S$ we mean in equation 3 as the number
$r=|s|$ and the matrix $\omega=|s|^{-1}s$. Then $s=r\omega$, and the
expression for $\nu$ in polar coordinates has the form

$d\nu(s)=r^{-1}\,dr\,d\omega,$

where $d\omega$ is the invariant measure on the sphere.

The following assertion can be checked verified directly.

######  Theorem 3.

The representation $\pi$ of the group $P$ in the Hilbert space $L^{2}(S,\nu)$,
where $d\nu(s)=|s|^{-4}ds$, is special and has a nontrivial cocycle of the
form

$b(g)=T(g)f-f,\quad\text{where }\,f(s)=e^{-|s|/2}.$ (4)

###  2.5 Extending the special nonunitary representation of the subgroup $P$
to the whole group $U(2,2)$

It remains to check that the special representation can be extended to a
nonunitary representation of the whole group $U(2,2)$. The construction of the
required extension is based on the following property of the group $U(2,2)$.
Every element $g\in U(2,2)$ can be uniquely written as a product $g=pk$, $p\in
P$, $k\in K$, where $K$ is the maximal compact subgroup, which consists of the
elements $k\in U(2,2)$ satisfying the relation $kk^{*}=e$, i.e., the subgroup
of block matrices of the form

$k=\left(\begin{array}[]{cc}\alpha&\beta\\\ \beta&\alpha\\\
\end{array}\right)$

where $\alpha\alpha^{*}+\beta\beta^{*}=e_{p}$ and
$\alpha\beta^{*}+\beta\alpha^{*}=0$ (the Iwasawa decomposition).

Let $T$ be the special representation of the subgroup $P$ in the Hilbert space
$H=L^{2}(S,\nu)$ defined in Theorem 2, and $b$ be the nontrivial cocycle
defined by (8). Denote by $H_{0}$ the linear invariant subspace in $H$ spanned
by the vectors $b(p)$, $p\in P$.

######  Lemma 3.

1. The subspace $H_{0}$ is dense in $H$.

2. The vectors $b(p)$, $p\neq e$, are linearly independent (the nondegeneracy property).

Let $T(k)$, $k\in K$, be the operators defined on the set of vectors $b(p)$,
$p\in P$, by the formula

$T(k)b(p)=b(p^{\prime}),$

where $p^{\prime}\in P$ is defined by the relation $kp=p^{\prime}k^{\prime}$,
$k^{\prime}\in K$.

3. The operators $T(k)$ satisfy the group relation $T(k_{1}k_{2})b(p)=T(k_{1})T(k_{2})b(p)$ for any $k_{1},k_{2}\in K$ and $p\in P$, and hence generate a representation of the subgroup $K$ in the subspace $H_{0}$.

######  Theorem 4.

The operators $T(k)$, $k\in K$, together with the operators $T(p)$, $p\in P$,
generate a representation of the whole group $U(2,2)$ in the subspace $H_{0}$,
in which the operator corresponding to the involution
$\sigma=\left(\begin{array}[]{cc}0&e_{p}\\\ e_{p}&0\\\ \end{array}\right)$ is
defined on the set of vectors $b(p)$ by the following formula:

$T(\sigma)b(p)=b(\hat{p}),\qquad\hat{p}\in P,$ (5)

where $\hat{p}$ is uniquely determined by the relation
$\hat{p}\hat{p}^{*}=\sigma pp^{*}\sigma.$ In this extension, the operators
corresponding to the elements of the subgroup $P$ are unitary, the operators
corresponding to the elements of the subgroup $K$ are bounded, and the
operator corresponding to the involution is unbounded and cannot be extended
to the whole space $H$. The extension of the $1$-cocycle $b$ from the group
$P$ to the group $U(2,2)$ is given by the formula

$b(pq)=b(p)\mbox{ for any }p\in P\mbox{ and }q\in Q.$ (6)

Applications of the described constructions to representations of the group of
currents will be described elsewhere. In this paper we restrict ourselves to
the group $U(2,2)$ just for methodological reasons: our aim is to give a
simple example of the general theory, which, in the authors’ opinion, covers a
wide class of semisimple Lie groups and the corresponding groups of currents.

## References

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Translated by N.Tsilevich.