# SEER-ZSL: Semantic Encoder-Enhanced Representations for Generalized Zero-
Shot Learning

William Heyden , Habib Ullah, M. Salman Siddiqui, Fadi Al Machot

Faculty of Science and Technology (REALTEK)
Norwegian University of Life Sciences
NMBU
1430 Ås, Norway
Corresponding author: William Heyden (e-mail<EMAIL_ADDRESS>
###### Abstract

Generalized Zero-Shot Learning (GZSL) recognizes unseen classes by
transferring knowledge from the seen classes, depending on the inherent
interactions between visual and semantic data. However, the discrepancy
between well-prepared training data and unpredictable real-world test
scenarios remains a significant challenge. This paper introduces a dual
strategy to address the generalization gap. Firstly, we incorporate semantic
information through an innovative encoder. This encoder effectively integrates
class-specific semantic information by targeting the performance disparity,
enhancing the produced features to enrich the semantic space for class-
specific attributes. Secondly, we refine our generative capabilities using a
novel compositional loss function. This approach generates discriminative
classes, effectively classifying both seen and unseen classes. In addition, we
extend the exploitation of the learned latent space by utilizing controlled
semantic inputs, ensuring the robustness of the model in varying environments.
This approach yields a model that outperforms the state-of-the-art models in
terms of both generalization and diverse settings, notably without requiring
hyperparameter tuning or domain-specific adaptations. We also propose a set of
novel evaluation metrics to provide a more detailed assessment of the
reliability and reproducibility of the results. The complete code is made
available on https://github.com/william-heyden/SEER-ZeroShotLearning/.

##  1 Introduction

Zero-shot learning (ZSL) stands at the forefront of machine learning and
computer vision challenges, pushing the boundaries of what AI systems can
achieve. It addresses the fundamental problem of recognizing and classifying
objects never seen or encountered during the training phase.

Figure 1: Our Semantic Encoder-Enhanced Representation (SEER) model is
designed to tackle the challenges posed by the inherent variances in semantic
data. It leverages the semantic space available from diverse sources, such as
word2vector (W2V), logic-explained network features (LEN), hand-annotated
semantics (SA), or other pertinent resources. This enables the extraction of
an adaptable latent space across multiple visual and semantic features,
covering both seen and unseen data. Our approach focuses on gaining precise
control over semantic information within these latent spaces while minimizing
the presence of non-relevant semantic details in the domain.

Recognizing unseen classes contains many practical scenarios across domains,
such as identifying unknown species, discerning unfamiliar material
compositions, or addressing limitations in data collection [1]. Zero-shot
learning accomplishes this through the transfer of shared knowledge of the
domain to infer previously unknown areas of the data manifold [2]. Hence, it
is evident that the compatibility between shared knowledge and the domain
plays a pivotal role in achieving high-performance zero-shot learning.

Traditional methods relied on matching visual attributes to object categories,
but this often led to a semantic gap [3]. More modern approaches leverage deep
learning to mitigate this gap by learning a joint embedding space of visual
and semantic features, enabling recognition of unseen categories through
methods like nearest-neighbour classification [4, 5, 6, 7, 8]. As ZSL evolves,
the focus remains on creating models capable of generalizing beyond seen
examples with improved accuracy. Building on these foundations, the advent of
latent space techniques in ZSL has received significant interest due to their
capacity to synthesize and interpret complex data distributions. These methods
offer a promising avenue for overcoming the challenges associated with the
semantic gap and improving generalization [9]. By utilizing the potential of
latent representations, researchers developed more sophisticated models that
effectively bridge the gap between seen and unseen classes, as demonstrated by
recent studies [10, 11, 12]. Other approaches aim to address these challenges
by leveraging the advancements made in generative models by creating pseudo-
data for potentially unseen instances. This data is then used within a simple
supervised learning paradigm, leading to significant success in benchmark
tests.

However, we argue that a disparity exists between how shared domain knowledge
is facilitated in such benchmark settings and what can be expected in more
realistic scenarios [1]. The commonly used shared knowledge comprises semantic
spaces covering both seen and unseen instances. Unfortunately, these semantic
spaces are often meticulously crafted for specific purposes, posing
limitations when applied in more realistic scenarios [13]. To address this
limitation, our paper introduces a novel approach emphasizing the crucial role
of generalization capabilities in generative-based zero-shot learning. The
central tenet of this approach revolves around achieving fine-grained control
over semantic information while mitigating the unintended presence of
semantic-unrelated information for the domain. We can also effectively steer
the generative model by augmenting the existing domain knowledge. The
generator can synthesize distinctive visual features and precise attribute
prototypes as both the visual and shared spaces reside within the same data
manifold. This dual strategy enhances the differentiation between classes and
addresses the core problems of bias and hubness that have hindered previous
ZSL methodologies. Hence, our approach not only alleviates the challenges of
dispersing semantic knowledge but also overcomes the generalization
capabilities necessary for advanced ZSL applications, thus setting new
benchmarks for AI systems in recognizing unseen objects. To summarize, our
main contributions of this paper are (1) We introduce a novel methodology that
enhances the state-of-the-art by achieving better generalization across a
spectrum of semantic spaces, reducing dependency on a specific space. We
provide a theoretical justification of the model supported by experimental
evidence. (2) We empirically demonstrate semantic space’s influence on
generative approaches for synthesizing previously unseen features. We further
substantiate these findings by evaluating the performance in different
settings of zero-shot. (3) We critically examine limitations and weaknesses
associated with the current settings in which generative approaches are
applied. Consideration of challenges related to the current reporting of
methodologies, particularly in terms of accuracy metrics, (4) We use a novel
semantic space based on the Logic Explained Networks (LEN) [14].

##  2 Related works

Figure 2: We simultaneously train (a) an encoder to extract unbiased semantic
information while prompting a generator to utilize the learned data
distribution inherent from the unknown probability space. (b) To guide the
generator, we introduce a classification loss that encourages the development
of high-level discriminative features. Subsequently, we optimize a conditional
variational encoder to align the extracted knowledge from the latent data
distribution conditioned on the true distribution (c). This methodology
enables classification in a SEER space using a simple interpretable distance
metric.

We survey the latest methods by categorizing them based on their approach to
latent space manipulation.

Contrastive Learning Approaches: These innovative methods enhance latent space
learning by contrasting data with semantic embeddings, thus improving the link
between seen and unseen classes. Examples include the Transferable Contrastive
Network [12], which ensures consistent visual-semantic alignment through
knowledge transfer, and the hybrid GZSL framework [11], which combines
generative and embedding models for improved feature mapping. However,
generalizing to unseen classes is a key challenge, as these methods need to
adapt learned representations from seen to unseen classes without bias.
Achieving a balance in visual-semantic alignment is also challenging requiring
careful hyperparameters calibration.

Disentanglement-Based Methods: Techniques like Generative-Based
Disentanglement and Conditional VAE with a Total Correlation (TC) module focus
on separating overlapping factors in the latent space for clearer, more
transferable representations [15, 6]. These methods aim to isolate meaningful
features in data but sometimes struggle to filter out noise well, leading to
less accurate generalization to unseen classes.

Attention Mechanisms and Hierarchical Models Attention-based methods, notably
in compositional ZSL, employ strategies like cross-attention for targeted
attribute analysis [16]. Hierarchical models [17] improve feature separation
through layered disentanglement. Attention mechanisms and hierarchical models
in zero-shot learning face challenges such as attribute imbalance, difficulty
in transferring nonsemantic features, and reliance on high-quality, detailed
attributes. These issues can impact their effectiveness and generalization to
unseen classes.

Cross-Modal and Optimization Techniques: Models that integrate cross-modal
latent spaces and latent space optimization techniques focus on aligning
attributes across different modalities and enhancing generalization to unseen
classes [18, 19]. Cross-modal and optimization techniques, which involve
integrating latent spaces across different modalities and optimizing these
spaces, face challenges in maintaining consistent attribute representation
across modalities, ensuring that attributes are uniformly and accurately
represented in each modality.

This paper merges the strengths of existing methods while addressing their
challenges. Therefore, we introduce four major contributions:

  * •

Improved Semantic Control: Generating latent features closely aligned with
semantic attributes ensures enhanced class differentiation, which is crucial
for class inference.

  * •

Noise Mitigation: Our method generates cleaner, class-specific features by
focusing on noise filtering, leading to more accurate classification of unseen
data.

  * •

Generalization and Bias Reduction: Enhanced control over the feature space
allows our model to overcome biases towards seen classes and address the
hubness problem.

  * •

Pseudo Prototype Generation: The precise semantic alignment enables the
generation of accurate attribute prototypes, aiding in the property inference
of unseen classes.

##  3 Methodology

Zero-shot learning (ZSL) aims to develop a predictive model capable of
understanding visual and semantic indicators to classify unseen categories-
essentially, generative ZSL becomes essential when there are no labeled
examples for all classes in question. Consequently, the dataset is divided
into a training set with seen classes denoted as
$\mathrm{Y}{seen}={y^{1}_{seen},y^{2}_{seen},\ldots,y^{n}_{seen}}$, and a
testing set with unseen classes represented as
$\mathrm{Y}{unseen}={y^{1}_{unseen},y^{2}_{unseen},\ldots,y^{n}_{unseen}}$. It
is critical to ensure that $\mathrm{Y}_{seen}\cap\mathrm{Y}_{unseen}=\phi$.
The challenge lies in constructing a function
$\mathbb{R}^{d}\rightarrow\mathrm{Y}_{unseen}$, which learns to predict the
unseen classes using the training set and subsequently tests its efficacy on
the unseen class data, maintaining the condition
$\mathrm{Y}_{seen}\cap\mathrm{Y}_{unseen}=\phi$. ZSL aspires to mimic human
adaptability to novel scenarios by leveraging models that can anticipate
without prior examples. Given a semantic $s\in S_{\text{seen}}$ and visual
$x\in X_{seen}$ space corresponding to seen classes, we first train a
Variational Autoencoder (VAE) [20] to learn a latent representation that is
informative of the semantic attributes. The VAE is trained with the standard
reconstruction loss and a Kullback-Leibler (KL) divergence loss:

$\mathcal{L}_{\text{VAE}}=\mathcal{L}_{\text{recon}}(s,\hat{s})+\beta\mathcal{L}_{\text{KL}}(\mathcal{N}(\mu(z),\sigma^{2})),$
(1)

where $\mathcal{L}_{\text{recon}}$ is the reconstruction loss,
$\mathcal{L}_{\text{KL}}$ is the KL divergence loss, $z$ is the latent
variable, $\hat{s}$ is the pseudo semantics, and $\beta$ is a weighting
coefficient. Exploiting a reconstruction loss enables effective generalization
to various data manifolds within the semantic space. Furthermore, the KL loss
imposes constraints on the spread of the generated latent space. The loss of
Eq. 1 ensures that the latent representation learned by the VAE is informative
and denoised of the original semantic attributes while retaining key
characteristics of the true data distribution $\mathbb{P}_{r}$. We assume the
semantic space $S$ are distributed closely related data manifold
$\mathbb{P}_{r}$ of the joint space
$\mathbb{E}_{s\sim\mathbb{P}(x,s)}s=\int[xp(s|x)]$ according to domain
adoption theory [21]. By introducing the adjustable parameter $\beta$, we can
balance the latent capacity of semantic information captured from the true
data manifold distribution $\mathbb{P}_{r}$. For a continuous semantic
distribution, the encoder parameterized by $\theta$ can be exploited to retain
certain properties $\mathbb{E}_{z\sim\mathbb{P}_{r}}[p_{\theta}(z|s)]$ where
feature-wise errors are minimized [22]. Assuming a continuous reconstructed
space that is reconstructed through the decoder $\hat{s}\sim q_{\theta}(s|z)$,
we ensure $\hat{s}$ captures the data distribution [23], while offering an
invariance towards manipulation, often referred to as style error [24].

Next, we utilize the latent space of the VAE as input to a Generative
Adversarial Network (GAN) [25]. A generator $g$, which is trained with a
Wasserstein loss with gradient penalty ($\mathcal{L}_{\text{Wasserstein-GP}}$)
and a class-label conditioning loss ($L_{\text{cls}}$):

$\mathcal{L}_{\text{GAN}}=\mathcal{L}_{\text{Wasserstein-
GP}}(g(z),X_{\text{seen}})+\eta L_{\text{cls}},$ (2)

where $X_{\text{seen}}$ are the visual features of seen classes.

Let $g:z\rightarrow X$ be a function of nonlinear affine transformation, then
$g$ sustains the prior $z$ to be contained in $\mathbb{P}_{r}$ [26]. To
address the instability of discriminators caused by potential disjoint support
when $dim(z)<dim(X)$ [27], we propose a criterion aiming to partially align
the two manifolds. Conditioning on a class-label loss $L_{\text{cls}}$ defined
in Eq. 3 $g(z)$ captures high-level features of the data manifold distribution
satisfying the properties of alignment. Moreover, the theory of low-
dimensional manifolds suggests that images existing on an $n$-dimensional
manifold can be condensed into a lower-dimensional space [27]. This concept
enables us to directly integrate semantic information into the generation
process, furhter ensuring alignment between the two manifolds within a
specific segment.

$L_{\text{cls}}=-\sum_{i}\sum_{C}\log[Pr(Y_{C}^{i}|X_{\text{seen}}^{i})],$ (3)

Guiding the generator by classification loss, with $Y_{C}^{i}$ representing
the true class label of the $i$-th sample and $Pr(Y_{C}^{i}|x^{i})$ being the
predicted probability, mitigates a perfect discriminator $D$, ensuring more
manageable gradients for the generator to enhance its performance.

The discriminator $D$ takes as input the output of the generator
$X_{\text{pseudo}}$ and $X_{\text{seen}}$, aiming to distinguish between the
generated and real samples. The overall loss for the enhanced embedding is:

$\mathcal{L}_{\text{emb}}=\mathcal{L}_{\text{GAN}}+\alpha\mathcal{L}_{\text{VAE}},$
(4)

Where $\alpha$ is a hyperparameter that balances the contribution of the VAE
loss in the semantic encoding loss function.

To ensure alignment between the enriched latent representations extracted from
the true data manifold and concurrently minimize biases towards the available
training information, our subsequent step involves training a conditional
variational autoencoder (CVAE) [5]. This CVAE is employed to strategically
situate and align these spaces within our framework. Additionally, evidence
shows that concatenating the pseudo visual data $g:z\rightarrow
X_{\text{pseudo}}$ and semantic data space $S_{\text{seen}}$ of the seen
classes further enhance the visual space for classification [28].

$X^{\prime}=X_{\text{pseudo}}\oplus S_{seen};$ (5)

To retain the distribution dependency of the manifolds, we consider
conditional mapping into the semantic encoded enhanced representation (SEER)
space. Without loss of generality, we assume an algebraic homomorphism of the
mapping [29]. The alignment of the learned representations can then be
achieved by conditioning on the respective spaces with a reconstruction loss
using our true data distribution [30]. Our loss function of the CVAE is then
given as

$\mathcal{L}_{\text{CVAE}}=\mathcal{L}_{\text{recon}}(x^{\prime},s)+\gamma\mathcal{L}_{\text{KL}}(z,\mathcal{N}(0,I)),$
(6)

Integrating $\gamma$ ensures that the manifested space are retaining
information from the true domain probability distribution in a suitable manner
for spatial classification. This explicitly generated enhanced space will
retain similar properties regardless of semantic information. Specifically,
the SEER provides properties of the probability space for a generalisation
over classes
$\mathbb{E}_{z\sim\mathbb{P}_{r}}=ln[p_{\theta}(z|x^{\prime},s)]\forall i\in
Y_{i}$ where $x^{\prime}\in X^{\prime}$.

For the final classification, we employ cosine similarity in the SEER space.
By assigning the positional label in the SEER space to the transformed source
data, we aim to reduce class divergence within the manifold by leveraging its
surjective properties [30]. Furthermore, this approach enables the data to
inherit the learned dependencies in the manifold structures, leveraging the
modality invariant distribution, resulting in greater generalization. The
class of a given sample $x_{unseen}$ is then determined based on the
minimization of cosine similarity of the embedding of a sample $z^{\prime}$:

$\begin{split}y(x_{unseen})=min\\{\cos[p_{\theta}(z^{\prime}|x_{unseen},s],z)\\}\\\
=\\{1-\frac{\sum_{i}^{n}(p_{\theta}(z_{i}|x_{unseen}),z)}{z\sqrt{\sum_{i}z_{i}^{2}}}\\}\forall
i\in Y\end{split}$ (7)

##  4 Dataset

We used three diverse benchmark datasets for visual representation: the SUN
Attribute (SUN) dataset [31], the CUB-200-2011 Bird (CUB) dataset [32], and
the AwA2 dataset [33]. The SUN dataset features 14,340 images of different
sceneries across 717 classes, distinguishing between 645 seen and 72 unseen
classes. The CUB dataset comprises 11,788 images of bird species; among 200
avian species, divided into 150 seen and 50 unseen classes. The AwA2 dataset
presents 37,322 images of animals, each with 40 seen and 10 unseen classes.
Notably, the SUN dataset contains the broadest range of semantic differences.
The CUB dataset annotates the finest granularity of 312 attributes indicating
various visual traits. The AwA2 dataset contains the most distinctive
features. The visual features are extracted from the ResNet101 backbone,
pretraind on ImageNet [13]. For a fair comparison of semantic enhancement, we
only compare with models using the same image features. To prove the effective
generalisation capability of our model, we include different semantic spaces
for these visual representations. These comprise of the word2vec [34], visual
grounded semantics annotations [35], and LEN logic-based rules (see Section
6). Hence, these collections encompass vastly different representations and
therefore work as an excellent proxy for real-world settings. We presume that
the true joint distribution is identical without loss of generalization.
Consequently, these datasets can be used to measure a model’s capability of
generating a concurrent SEER space, which directly correlates with its
generalisation ability.

##  5 Experimental results

Following prior work in zero-shot classification, we report generalized top-1
accuracy considering the main benchmark datasets. The average per-class
accuracy is determined as follows:

$acc^{class}_{average}=\frac{1}{|Y|}\sum_{i=0}^{|Y|}\left(\frac{N_{correct_{class}}^{class_{i}}}{N^{class_{i}}_{Total}}\right)$
(8)

| AWA2 |  |  | CUB |  |  | SUN |  |
---|---|---|---|---|---|---|---|---|---
Methods | S | U | H | S | U | H | S | U | H
CVAE-GZSL (18) [5] | - | - | 51.2 | - | - | 34.5 | - | - | 26.7
cycle-CLSWGAN* (18) [36] | 64.0 | 56.9 | 60.2 | 61.0 | 45.7 | 52.3 | 33.6 | 49.4 | 40.0
f-CLSWGAN* (18) [37] | 61.4 | 57.9 | 59.6 | 57.5 | 43.7 | 49.7 | - | - | 39.4
f-VAEGAN-D2 (19) [38] | 70.6 | 57.6 | 63.5 | 60.1 | 48.4 | 53.6 | 38.0 | 45.1 | 41.3
LisGAN* (19) [39] | 76.3 | 52.6 | 62.3 | 57.9 | 46.5 | 51.6 | 37.8 | 42.9 | 40.2
CADA-VAE (19) [40] | 75.0 | 55.8 | 63.9 | 53.5 | 51.6 | 52.4 | 35.7 | 47.2 | 40.6
GDAN (19) [41] | 67.5 | 32.1 | 43.5 | 66.7 | 39.3 | 49.5 | 89.9 | 38.1 | 53.4
DAZLE (20) [42] | 75.7 | 60.3 | 67.1 | 59.6 | 56.7 | 58.1 | 24.3 | 52.3 | 24.3
IZF (20) [43] | 77.5 | 60.6 | 68.0 | 68.0 | 52.7 | 59.4 | 57.0 | 52.7 | 54.8
SDGZSL (21) [6] | 73.6 | 64.6 | 68.8 | 66.4 | 59.9 | 63.0 | - | - | -
DAA-GZSL (23) [44] | 79.9 | 65.7 | 72.1 | 65.5 | 66.1 | 65.8 | 38.7 | 47.8 | 42.8
SE-GZSL (23) [45] | 68.1 | 58.3 | 62.8 | 53.3 | 41.5 | 46.7 | 30.5 | 50.9 | 34.9
SEER-ZSL (Ours) | 90.3 | 59.2 | 71.5 | 89.6 | 50.8 | 64.8 | 71.2 | 56.7 | 63.1
Table 1: Performance of our proposed SEER space in the generalized zero-shot
setting over state-of-the-art ZSL models. Using the proposed split for
unseen/seen classes per dataset, including 20% seen classes in the generalized
setting. The result is reported using transductive assumption for seen (S),
unseen (U), and harmonic mean (H= $[\frac{2*S*U}{S+U}]$). * indicates
benchmark agains AwA1 as AwA2 is not reported.

###  5.1 Implementation details

Our VAE and CVAE networks use a symmetric encoder-decoder MLP, compressing
latent space $z$ to 48 dimensions for VAE and $d$ dimensions for SEER, where
$d$ represents the semantic dimensions. ReLU activation [46] and a 0.5 dropout
rate are employed for non-linearity and regularization. The WGAN’s generator
has three MLP layers (215, 516, 1024) and a 2048-dimensional output with a
symmetric discriminator. To addresses the instability when sampling
$z\sim\mathbb{P}_{r}$ we adopt tight training of the generator [47]. Training
involves recursive updates every 100 iterations with a 0.001 learning rate and
exponential decay. The model, built in PyTorch, is trained on an NVIDIA RTX
8000 (48GB) for 4-6 hours, varying by dataset size.

###  5.2 Generalized accuracy

For fair comparisons, the hyper-parameters are shared across the datasets,
excluding $\beta$ and $\gamma$ of the loss functions. These parameters balance
the activation of latent semantic probability distributions within their
respective data manifolds. The results in Table 1 show how our SEER space
effectively closes the accuracy gap across different datasets, yielding a
harmonic mean range of approximately 7% between AWA2 and SUN and 6% between
SUN and CUB using the same hyper-parameters. This contrasts to tuned state-of-
the-art models, which exhibit harmonic disparities of around 15% between AWA2
and SUN and around 20% between SUN and CUB. We note that the higher harmonic
mean is related to generalisation abilities and is not biased toward the seen
or unseen domain. Considering the SUN dataset, we achieve a harmonic mean
improvement of 10% compared to other models. The diverse range of classes in
the SUN dataset presents a significant challenge in generalization. However,
our model addresses it effectively, as the performance metrics show. This
highlights our model’s advanced semantic processing and robust generalization
ability, which is crucial in GZSL. Furthermore, models excelling in one
domain, such as AWA2, often encounter reduced efficiency when confronted with
shifts in the granularity of the data manifold, as in SUN. This phenomenon is
known as the semantic shift problem [30]. For unseen classes exclusively, we
experience at most under 8% drop in accuracy from AWA2 to CUB, while the range
for most models lies around 10%. Additionally, our approach consistently
outperforms existing state-of-the-art models in absolute accuracy
measurements.

Figure 3: To assess and comprehend the generalization capabilities of the
proposed model based on semantic information, we adopt SEER across various
semantic distributions. This violin figure illustrates the summary statistics
of the semantic space. The variation in smoothness indicates the data density
for the different semantic spaces.

###  5.3 Semantic generalization

The enhanced semantic embedding accomplished by the encoder enables the
generator to synthesize visual features that align with the true probability
distribution within the data manifold. Consequently, we can refine our mapping
of the visual-semantic space to our SEER space, given they originate from the
same data distribution. The efficiency of this dual approach is evident in
Fig. 4. This is constructed without any change in the hyper-parameters. The
linear variance across dimensions in our SEER space ensures comprehensive
coverage of maximum variability within the distribution. This stands in
contrast to alternative semantic spaces, where a few dimensions tend to
encompass a significant portion of available information. Consequently,
exclusive distinguishing classes based on this distribution becomes
exceedingly challenging in higher dimensions [48]. This mitigation strategy
extends to unseen classes, facilitated by an encoder guided through an
impartial reconstruction loss. However, some discriminative information from
the unseen distribution might be compromised due to the influence of the KL
divergence [49]. Empirical evidence demonstrates that tuning the $\beta$
parameters governing this balance yields consistently strong results. We
hypothesize that this outcome arises partially because the generator can
compensate for the loss of information by generating features from the
simpler, yet contained in the true probability distribution [44].

Figure 4: The uniform variance across dimensions using Principle Components
Analysis (PCA) within our SEER space displays a comprehensive representation
of the true distribution. Compared to semantic space from other domain, the
uniform variance ensures smoother generative capabilities from this latent
space.

Fig. 3 illustrates the significant disparities which can be observed in the
auxiliary information. The figure displays the average class-wise distribution
of the feature space. As depicted, the original hand-annotated semantic space
[33] offers extensive coverage and distribution across the semantic manifold.
This starkly contrasts with alternative representations of auxiliary
information. The designed semantic information fails to represent a
universally applicable expectation of information, thereby imposing
limitations on the robustness and transferability of a zero-shot model. To
address this limitation, we apply our model in additional test environments
that reflect a more realistic and diverse distribution where zero-shot
learning can be implemented in real-world scenarios. Specifically, we report
accuracy on word2vector (W2V) data generated from the web [34] and visually
grounded semantic annotations (VGSE) [35]. The impact of semantic space in
generative approaches to zero-shot learning becomes apparent in the decline in
accuracy observed in empirical experiments, as presented in Table 3. Comparing
Table 1 with Table 3, it is evident that the employed models exhibit a notable
reduction in harmonic accuracy, ranging from 15% to 20% depending on the
semantic space. Furthermore, the decrease in accuracy is considerably more
pronounced in the semantic spaces characterized by the narrowest distributions
(see Fig. 3).

Classification space | Transductive | Inductive
---|---|---
Visual | 58.48 | 29.19
SEER | 69.13 | 47.88
Table 2: Accuracy for AwA in different zero shot settings.

###  5.4 Validity metrics

To comprehensively assess the performance of our model, we further examined
its accuracy across various measures and zero-shot conditions. We consider a
cross-examination of a total of four settings. In the inductive setting, our
training assumes access solely to labeled data from the seen classes. However,
in the transductive setting [50], we additionally assume access to the
unlabeled test inputs from the unseen semantic space. This results in more
information from the true data manifold being accessible during training.
Despite this information being unlabeled, our model shows evidence of enhanced
exploitation of this knowledge, resulting in significantly increased scores in
this specific setting, around 25% for visual space and around 20% for semantic
space, as shown in Table 2. Furthermore, we validate the SEER space’s
generalization ability by assessing its classification accuracy for both
visual and semantic unseen labels. Due to our chosen classification method of
nearest neighbor search, we expect the semantic space to perform better,
primarily due to the curse of dimensionality in the visual manifold. Despite
this expectation, our model achieves performance comparable to visual
embedding models. For a comprehensive list of visual embedding models, refer
to [28].

|  | AWA |  |  |  | CUB |  |  |  | SUN |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Space | Method | Conv. | Gen. |  |  | Conv. | Gen. |  |  | Conv. | Gen. |  |
|  | H | S | U | H | H | S | U | H | H | S | U | H
W2V [34] | f-VAEGAN-D2 | 58,4 | 59 | 46,7 | 52,5 | 32,7 | 44,5 | 23 | 30,3 | 39,6 | 33,3 | 25,9 | 29,1
| CADA-VAE | 49,0 | 60,1 | 38,6 | 47 | 22,5 | 39,7 | 16,3 | 23,1 | 37,8 | 28,2 | 26 | 27
| Ours | 43,8 | 71,2 | 41,6 | 52,5 | 15,5 | 19,3 | 20,1 | 19,7 | 35,3 | 44,6 | 33,2 | 38,1
VGSE [35] | f-VAEGAN-D2 | 61,3 | 66,7 | 45,7 | 54,2 | 35 | 45,7 | 24,1 | 31,5 | 41,1 | 35,7 | 25,5 | 29,8
| CADA-VAE | 52,7 | 61,6 | 46,9 | 53,9 | 24,8 | 44,5 | 18,3 | 25,9 | 40,3 | 29,6 | 29,4 | 29,5
| Ours | 79,8 | 85,2 | 62,3 | 72,0 | 28,1 | 32,4 | 28,1 | 30,1 | 31,9 | 42,1 | 34,2 | 37,7
LEN | Ours | 78,2 | 89,3 | 56,7 | 69,4 | 64,4 | 86,1 | 33,18 | 47,9 | 41,6 | 52,4 | 52,2 | 52,3
Table 3: Accuracy of generalized zero-shot learning given different semantic
information.

To further justify the substantiation of the predictive quality and
information retrieval capacity of the SEER space, we incorporate the confusion
matrix detailing the classification within this specific space. Utilizing the
conventional computation formulas for recall ($\frac{tp}{(tp+fn)}$) and
precision ($\frac{tp}{(tp+fp)}$), we derive scores as presented in Table 4. As
observed in Table 4, our model demonstrates a slight bias toward capturing all
positive labels, resulting in higher accuracy for recalls. We attribute this
tendency to the generalization capabilities inherent in our SEER space, which
notably encompasses the true data manifold. Consequently, the enhanced encoded
representation efficiently mimics the true data distribution but potentially
overestimates specific areas. Considering the confusion matrix depicted in
Fig. 5, the overestimation emerges predominantly from seen classes, implying a
slight bias in our model. We argue that this bias stems from the trade-off
between the generalization capabilities of our model and its robustness in
recognizing diverse data manifolds. Empirically, evidence supporting this can
be found in misclassification of seen classes remaining in the seen domain.
This highlights how the modes of the class-wise distributions caused by
imbalances, rather than generalization, drive such behavior.

| Recall | Precision
---|---|---
AWA | 71.8 | 63.5
CUB | 70.0 | 57.2
SUN | 65.6 | 55.8
Table 4: Precision and recall accuracy. Figure 5: In the confusion matrix, each square on the diagonal represents the number of testing images classified correctly for each testing class in AWA2. The evenly distributed prediction shows that our model is able to generalize well across the probability space. Seen classes 6-49, and unseen classes 36-11. AWA | LEN
---|---
exc/ VAE | 23.8
exc/ WGAN | 33.9
exc/ CVAE | 40.0
exc/ VAE+WGAN | 25.5
Table 5: Result from ablation study.

##  6 Ablation study

In our ablation study, we explore the integration of Logic Explained Networks
(LENs) [14] within a Zero-Shot Learning (ZSL) framework to enhance both model
interpretability and performance. We focus on manipulating the semantic
embedding space, denoted as $S$, and mapping it to a set of classes $C$. The
LEN operates as a function $\text{LEN}:S\rightarrow C$, where each element
$s_{i}\in S$ is mapped to a class $c_{j}\in C$. This process is augmented by
applying LEN-generated conjunctive rules to the semantic features within $S$.
Specifically, for each semantic feature $f_{k}$ in the embedding $s_{i}$,
identified as relevant (non-negated) in the top-1 explanation provided by LEN,
we modify $s_{i}$ by doubling the values of $f_{k}$, represented as
$s^{\prime}_{i}=s_{i}\times 2$ for all relevant $f_{k}$ in $s_{i}$. This
amplification aims to enhance the discriminative power of critical features
without causing computational instability, as indicated by empirical studies.
Additionally, to address the challenge of single-sample representation in our
semantic space, we introduce white noise $\eta\sim N(0,1)$ to each sample in
$S$, subsequently generating 100 instances per class from these perturbed
samples. This enhanced embedding space $S^{\prime}$ is then utilized in the
ZSL model, enabling us to assess the impact of the LEN-induced explanations
and the resulting modifications in the embedding space.

In Table 5, a substantial drop of nearly 40% in model performance is observed
when excluding the VAE from the training setup (see Section 5). Despite the
LEN semantic space $S^{\prime}$ introducing an amplification of critical
regions of the data manifold, the model struggles with capturing a robust
semantically encoded enhanced representation (SEER) space in this setting.
Notably, excluding the WGAN from training results in a 10% increase in
accuracy, highlighting the potency of the semantic information contained
within the latent space $z$. The alignment of the dual manifold is
accomplished through the CVAE, as shown by a 29% accuracy drop from the
original implementation, highlighting its effectiveness. This is further
substantiated by the significant decrease when enforcing an uninformative
manifold by excluding both the VAE and the WGAN. Table 5 illustrates the
efficient nature of our dual learning structure to generalize well from the
complex interplay in our data distribution.

##  7 Conclusion

In this paper, we have presented a novel method aimed at establishing a more
universally applicable approach to zero-shot learning. Our experiments reveal
that existing state-of-the-art approaches struggle to generalize effectively
across broader and more realistic settings. To address this limitation, we
introduced a robust and transformation-invariant classification space called
SEER (Semantic Encoded Enhanced Representation). We have demonstrated that
utilizing this space enables achieving higher accuracy in classification
across various data manifolds. Future endeavors will focus on fully leveraging
and exploring the advantages offered by this enhanced space. Promising
directions for further exploration involve enhancing generative capabilities,
not only concerning unseen classification but also fostering creativity and
detecting novelty within the visual manifold. This pursuit aims to push the
boundaries of innovation and creativity within the framework of zero-shot
learning.

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# Phase transitions for the geodesic flow of a rank one surface with
nonpositive curvature

K. Burns, J. Buzzi, T. Fisher, and N. Sawyer  K. Burns, Department of
Mathematics, Northwestern University, Evanston, IL 60208, USA, _E-mail
address:_<EMAIL_ADDRESS>J. Buzzi, Laboratoire de Mathématiques
d’Orsay, CNRS - UMR 8628 Université Paris-Sud 11, 91405 Orsay, France, _E-mail
address:_<EMAIL_ADDRESS>T. Fisher, Department of Mathematics,
Brigham Young University, Provo, UT 84602, USA, _E-mail address:_
<EMAIL_ADDRESS>N. Sawyer, Department of Mathematics and Computer
Science, Southwestern University, Georgetown, TX 78626, USA, _E-mail address:_
<EMAIL_ADDRESS>
###### Abstract.

We study the one parameter family of potential functions $q\varphi^{u}$
associated with the geometric potential $\varphi^{u}$ for the geodesic flow of
a compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that
there is a unique equilibrium state associated with $q\varphi^{u}$, and it has
full support. For $q>1$ it is known that an invariant measure is an
equilibrium state if and only if it is supported on the singular set. We study
the critical value $q=1$ and show that the ergodic equilibrium states are
either the restriction to the regular set of the Liouville measure, or
measures supported on the singular set. In particular, when $q=1$, there is a
unique ergodic equilibrium state that gives positive measure to the regular
set.

###### Key words and phrases:

Dynamical systems; equilibrium states; geodesic flow

###### 2010 Mathematics Subject Classification:

37B40, 37C40, 37D30

T.F. is supported by Simons Foundation Grant # 239708.

J.B. is supported by the ANR grant ISDEEC # ANR-16-CE40-0013.

##  1\. Introduction

Thermodynamical formalism brings ideas from statistical mechanics into ergodic
theory [23, 25], and builds natural measures such as the Sinaĭ-Ruelle-Bowen
measures or the maximal entropy measures as _equilibrium states_ with respect
to appropriate _potential functions_. In the classical setting of uniformly
hyperbolic dynamics (e.g., geodesic flows on closed manifolds of negative
curvature) every Hölder continuous potential function defines a unique
equilibrium measure. This equilibrium state can be characterized in several
ways. It is the limit of uniform measures on the periodic orbits with weights
determined by the potential. Also the integral of a continuous function with
respect to the equilibrium state is the forward Birkhoff average of the
function at a Lebesgue typical point.

Since the 1980s, this formalism has been studied in various nonuniformly
hyperbolic settings. The geodesic flows for closed _rank one manifolds_ with
nonpositive curvature [1] have been key examples. New phenomena appear such as
_phase transitions_ : for special potential functions, the uniqueness of the
equilibrium state fails. A physical analogue is the coexistence of ice and
liquid water at a precise temperature for a given pressure [15, 24].

In this paper we consider the geodesic flow for a closed rank one surface with
nonempty singular set and the family of potentials $q\varphi^{u}$, where
$\varphi^{u}$ is the _unstable Jacobian potential_ , which we define below,
and $q\in\mathbb{R}$. It is known (see for example the introduction to [5])
that this family has a phase transition at $q=1$. Using the Ledrappier-Young
entropy theory, we show that at the phase transition, when $q=1$, the only
equilibrium state not carried by the singular subset is the regular Liouville
measure restricted to the regular set.

##  2\. Precise statement and previous results

We consider the dynamical systems defined by a flow
$\mathcal{F}=(f^{t})_{t\in\mathbb{R}}$ on a manifold and a potential
$\varphi$, which is a continuous real valued function on the manifold. The
topological pressure $P(\varphi)$ is defined as

$P(\varphi)=\sup_{\mu}h_{\mu}(\mathcal{F})+\int\varphi\,d\mu,$

where the supremum is over all $\mathcal{F}$-invariant Borel probability
measures. An _equilibrium state_ is a measure that achieves the supremum. By a
theorem of Newhouse [20] (see [6] for a simplified proof) we know that
equilibrium states always exist if $\mathcal{F}$ is $C^{\infty}$. We will
consider equilibrium states for the one parameter family of potentials
$q\varphi$ for $q\in\mathbb{R}$.

This paper studies a specific class of geometric examples. The flow
$\mathcal{F}$ is the geodesic flow on the unit tangent bundle $T^{1}M$ of a
compact connected $C^{\infty}$ Riemannian surface $M=(M,g)$. We assume that
the Gaussian curvature $K$ of the metric $g$ is everywhere nonpositive and
somewhere negative. Such surfaces are known as rank one surfaces of
nonpositive curvature; this is the two dimensional case of the notion of a
rank one manifold of nonpositive curvature introduced in [2]. As a consequence
of the Gauss-Bonnet theorem, $M$ has genus at least $2$.

In this setting, there are two continuous invariant subbundles $E^{s}$ and
$E^{u}$ of $TT^{1}M$. Each of these is one dimensional and orthogonal to the
flow direction $E^{c}$ in the natural Sasaki metric. As we explain in the next
section, $E^{s}$ corresponds to the orthogonal Jacobi fields whose length is
nonincreasing for all time and $E^{u}$ to the orthogonal Jacobi fields whose
length is nondecreasing for all time. A vector $v\in T^{1}M$ is said to be
_regular_ if $E^{s}(v)\cap E^{u}(v)=\\{0\\}$. Otherwise $v$ is said to be
_singular_. If $v$ is singular, $E^{s}(v)=E^{u}(v)$ and there are nonzero
covariantly constant orthogonal Jacobi fields along the geodesic $\gamma_{v}$
that has initial tangent vector $v$. It is easily seen that $v$ is singular if
and only if $K(\gamma_{v}(t))=0$ for all $t$. We denote by $\mathcal{R}$ the
set of all regular vectors and by $\mathcal{S}$ the set of all singular
vectors. The set $\mathcal{R}$ is obviously open and invariant while
$\mathcal{S}$ is closed and invariant.

There is a natural smooth measure on $T^{1}M$ known as the Liouville measure.
It is invariant under the geodesic flow. The set $\mathcal{R}$ was shown to be
dense by Ballmann [1]. It has full Liouville measure in all known examples,
but whether this is always true is a major open question. This set
$\mathcal{R}$ however always has positive Lebesgue measure so one can restrict
the Liouville measure to $\mathcal{R}$ and then renormalize it to be a
probability measure, which will be denoted by $\mu_{L}$.

The potential function that we study is the _geometric potential_

$\varphi^{u}(v)=-\lim_{t\to 0}\frac{1}{t}\log\det(df^{t}|_{E^{u}_{v}}).$

This potential has been studied in the context of hyperbolic systems since the
pioneering work of Bowen. In our setting, $\varphi^{u}=0$ on $\mathcal{S}$ and
$\varphi^{u}(v)<0$ for any $v\in\mathcal{R}$ such that $K(\gamma_{v}(t))<0$
for some $t<0$. In particular, $\varphi^{u}(v)<0$ if $v$ is generic for any
invariant probability measure $\mu$ such that $\mu(\mathcal{R})=1$; see Lemma
3.1.

We consider the one parameter family of potentials $q\varphi^{u}$ where $q$
ranges over $\mathbb{R}$.

######  Definition 2.1.

For a potential $\varphi$, a _phase transition_ occurs at $q_{0}$ if the map
$q\mapsto P(q\varphi)$ is not differentiable at $q_{0}$.

Note that nondifferentiability implies nonuniqueness of the equilibrium states
by a simple convexity argument; see [22].

If $\mathcal{S}=\emptyset$, the geodesic flow is transitive and Anosov [13].
Any multiple of the geometric potential $\varphi^{u}$ is Hölder continuous,
and it is a classical result of Bowen [3] that all Hölder continuous
potentials have unique equilibrium states when the flow is transitive and
Anosov. All of these equilibrium states are Bernoulli, fully supported and the
weak∗ limit of weighted measures supported on periodic closed orbits. So there
are no phase transitions in this case and we henceforth assume that
$\mathcal{S}\neq\emptyset$. Bowen’s results hold in any dimension.

If $\mathcal{S}\neq\emptyset$ and $M$ is a surface, then the geodesic flow
lies on the boundary of the Anosov flows. Phase transitions are known to arise
for some systems similarly on the boundary of hyperbolic systems. For
instance, the partially hyperbolic maps introduced in [11] that are on the
boundary of uniformly hyperbolic diffeomorphisms have a phase transition as
shown by Leplaideur, Oliveira, and Rios [19] for a family of potential
functions related to the geometric potential. Specifically, for a partially
hyperbolic diffeomorphism $F$ with center distribution $E^{c}$ they show that
the family of potential functions $\phi_{q}=q\log|DF|_{E^{c}}|$ has a phase
transition for a specific parameter $q_{0}$. For certain partially hyperbolic,
topologically transitive, local diffeomorphisms there are phase transitions
[12] where each ergodic measure has positive entropy.

When $\mathcal{S}\neq\emptyset$ the first major result was by Knieper who used
a combination of ideas from Patterson-Sullivan and Bowen to prove uniqueness
of the measure of maximal entropy [16]. Climenhaga and Thompson [10] developed
a nonuniform version of Bowen’s techniques, and this was used in [4] to show
that $q\varphi^{u}$ has a unique equilibrium state for any $q$ sufficiently
close to 0. As in the Anosov case, these equilibrium states are Bernoulli,
fully supported and the weak∗ limit of weighted measures supported on periodic
closed orbits. The Bernoulli property is a consequence of work of Ledrappier,
Lima and Sarig [18].

Stronger results hold for surfaces with $\mathcal{S}\neq\emptyset$. In this
case, Theorem C of [4] shows there is a unique equilibrium state for
$q\varphi^{u}$ whenever $q<1$; these measures have all of the properties
described in the previous paragraph.

The results in [4] also apply to some other potential functions. Climenhaga
and Thompson’s techniques have also been used to show there is a unique
measure of maximal entropy for geodesic flows on certain manifolds without
conjugate points [9], and to establish the uniqueness of equilibrium states
for certain diffeomorphisms with weak forms of hyperbolicity [7, 8].

The pressure of $q\varphi^{u}$ was also studied in [5] as part of an
investigation of the Hausdorff dimensions of the level sets of the Lyapunov
exponent associated to $E^{u}$. In that paper it was noted that there is a
phase transition at $q=1$. The graph of $P(q\varphi^{u})$ is shown in Figure
1, which is essentially the same as Figure 2a) in [5].

Figure 1. Graph of the pressure

As we explain in Section 3, the following facts follow from the variational
principle, Ruelle’s inequality and the Pesin entropy formula. First

(1) $P(q\varphi^{u})\geq h_{\mu_{L}}(\mathcal{F})+q\int\varphi^{u}\,d\mu_{L},$

and both sides are equal to $0$ when $q=1$.

For $q>1$, we have $P(q\varphi^{u})=0$ which is trivially $C^{1}$ on
$(1,\infty)$ and the equilibrium states for $q\varphi^{u}$ are the invariant
probability measures supported on the compact invariant set $\mathcal{S}$.

For $q<1$, Theorem C of [4] shows that $P(q\varphi^{u})$ is $C^{1}$ on
$(-\infty,1)$ and that there is a unique equilibrium state $\mu_{q}$ for
$q\varphi^{u}$. Moreover, this equilibrium state is ergodic since it is unique
and satisifies $\mu_{q}(\mathcal{R})=1$, since the regular set has positive
measure with respect to $\mu_{q}$ and the regular set is invariant.

For $q=1$, since the integral on the right hand side of (1) is positive, there
must be a discontinuity in the derivative at $q=1$. Hence, there is a phase
transition at $q=1$. Both the invariant probability measures supported on
$\mathcal{S}$ and $\mu_{L}$ are equilibrium states for $\varphi^{u}$. The main
result of this paper is that there are no other equilibrium states for this
value $q=1$:

######  Theorem 2.2.

For $\varphi^{u}$ the only ergodic equilibrium state not supported on
$\mathcal{S}$ for the geodesic flow of a compact rank 1 surface of nonpositive
curvature is $\mu_{L}$.

We note that little is known about phase transitions for the geodesic flow of
compact rank one manifolds with nonpositive sectional curvature in higher
dimension. The Heintze example [2, Example 4] shows that at least two
different behaviours are possible. Heintze’s example is constructed by taking
two copies of a compact hyperbolic manifold of dimension $n\geq 3$ with finite
volume and one cusp, cutting off the ends of the cusps, and gluing the two
halves together. The resulting manifold has nonpositive curvature and contains
a flat totally geodesically isometrically embedded torus $T$ of dimension
$n-1$. Moreover, it can be arranged that each plane spanned by a vector
orthogonal to $T$ and a vector tangent to $T$ has sectional curvature zero or
that all such planes have negative curvature. In the first case, we see the
same phase transition at $q=1$ as in the surface case. In the second case, as
explained in [4, Section 10.2], it can be arranged that $q\varphi^{u}$ has a
unique equilibrium state for each $q\in\mathbb{R}$ and there are no phase
transitions. On the other hand, Theorem D of [4] proves that the family
$q\varphi^{u}$ does not have a phase transition at $q=0$ for the geodesic flow
of any compact rank 1 manifold with nonpositive sectional curvature.

######  Question 2.3.

For any compact rank one manifold with nonpositive sectional curvature is
there a unique equilibrium state associated with $q\varphi^{u}$ for each
$0<q<1$? What about for $q<0$?

######  Question 2.4.

For a compact rank one manifold with nonpositive sectional curvature, what
conditions guarantee that a phase transition occurs at some
$q^{\prime}\in\mathbb{R}$ for the one parameter family of potential functions
$q\varphi^{u}$?

##  3\. Jacobi fields and the derivative of the geodesic flow

We outline some facts about the geodesic flow for manifolds with nonpositive
curvature. A convenient source for the details is [14]. If $M$ is a Riemannian
manifold and $\pi:TM\to M$ is the natural projection, we have for each $v\in
TM$ a natural isomorphism $I_{v}:T_{v}TM\to T_{\pi(v)}M\times T_{\pi(v)}M$
defined as follows. Suppose $V(t)$ is a $C^{1}$ curve in $TM$ with footpoint
curve $c(t)$ and $V^{\prime}(t)$ is the covariant derivative of $V$ along $c$.
Then

$I_{V(0)}(\dot{V}(0))=(\dot{c}(0),V^{\prime}(0)).$

We have $I_{v}(T_{v}T^{1}M)=T_{\pi(v)}M\times v^{\perp}$, where $v^{\perp}$ is
the orthogonal complement of $v$ in $T_{\pi(v)}M$. The vector field that
generates the geodesic flow has value $I_{v}^{-1}(v,0)$ at $v$. If
$\xi=I_{v}^{-1}(w,w^{\prime})\in T_{v}TM$, and $J_{\xi}$ is the Jacobi field
along the geodesic $\gamma_{v}$ with $J(0)=w$ and $J^{\prime}(0)=w^{\prime}$,
then

$I_{F^{t}(v)}(DF^{t}(\xi))=(J_{\xi}(t),J^{\prime}_{\xi}(t))$

for all $t$. The _Sasaki metric_ is defined on $TTM$ by

$\langle\xi,\xi\rangle=\langle w,w\rangle+\langle
w^{\prime},w^{\prime}\rangle\quad\text{ if $\xi=I_{v}(w,w^{\prime})$.}$

A Jacobi field $J(t)$ along the geodesic $\gamma_{v}$ is _stable_ (resp.
_unstable_) if $J(t)\perp\dot{\gamma}(t)$ for all $t$ and $\|J(t)\|$ is
nonincreasing (resp. nondecreasing) for all $t$. A stable or unstable Jacobi
field can vanish only if it is identically zero. If $w$ is a vector
perpendicular to $v$, there are a unique stable Jacobi field and a unique
unstable Jacobi field along $\gamma_{v}$ with initial value $w$. The spaces of
stable and unstable Jacobi fields along $\gamma_{v}$ are therefore both one
dimensional. They also vary continuously. Consequently there are continuous
functions $a^{s},a^{u}:TT^{1}M\to\mathbb{R}$ such that if $w\in v^{\perp}$
then the stable (resp. unstable) Jacobi field along $\gamma_{v}$ with initial
value $w$ has derivative $a^{s}(v)w$ (resp. $a^{u}(v)w$) at time $0$. It is
clear that $a^{u}\geq 0\geq a^{s}$.

We set

$E^{s}(v)=\\{\xi\in T_{v}T^{1}M\text{ such that $J_{\xi}$ is stable}\\}$

and

$E^{u}(v)=\\{\xi\in T_{v}T^{1}M\text{ such that $J_{\xi}$ is unstable}\\}.$

It follows from the Jacobi equation that $\langle J,J\rangle$ is a convex
function for any Jacobi field $J$ along any geodesic $\gamma$. Moreover
$\langle J,J\rangle^{\prime\prime}(t)>0$ unless $J(t)=0$ or $K(\gamma(t))=0$.
It follows that stable and unstable Jacobi fields must have constant length
and be covariantly constant along singular geodesics because the curvature is
always zero. If $J$ is a nonzero unstable (resp. stable) Jacobi field and
$K(\gamma(t_{0}))<0$, then $\|J\|^{\prime}(t)>0$ for all $t>t_{0}$ (resp.
$\|J\|^{\prime}(t)<0$ for all $t<t_{0}$). We see that $\|J\|$ cannot be
constant if $J$ is a stable or unstable Jacobi field along a regular geodesic.
It is also clear that $a^{u}(v)=0=a^{s}(v)$ if $v$ is singular and
$a^{u}(v)>0>a^{s}(v)$ if the curvature is negative at $\pi(v)$.

The functions $a^{s}$ and $a^{u}$ are bounded because they are continuous. It
follows easily that if $\xi\in E^{s}(v)$ or $\xi\in E^{u}(v)$ and
$\lim_{|t|\to\infty}\frac{1}{t}\log\|Df^{t}(\xi)\|$ exists, then it is the
same as $\lim_{|t|\to\infty}\frac{1}{t}\log\|J_{\xi}(t)\|$. From this it is
easy to see that all three Lyapunov exponents for the geodesic flow
$\mathcal{F}$ are $0$ on the singular set, while on the regular set we have
the zero exponent associated to the flow direction and the exponents
$\chi^{u}$ and $\chi^{s}$ associated to the bundles $E^{u}$ and $E^{s}$. It is
obvious from the above discussion that if $v$ is regular and has all three
Lyapunov exponents defined, then $\chi^{u}(v)\geq 0\geq\chi^{s}(v)$.

######  Lemma 3.1.

Let $\mu$ be an ergodic probability measure such that $\mu(\mathcal{R})=1$.
Then $\chi^{u}(v)>0>\chi^{s}(v)$ for $\mu$-a.e. $v$.

###### Proof.

We must have

$\int a^{u}(v)\,d\mu(v)>0.$

Otherwise, since $a^{u}\geq 0$, we would know that $\mu$-a.e.
$v\in\mathcal{R}$ has $a^{u}(f^{t}(v))=0$ for almost all $t$. Since $a^{u}$ is
continuous, this would mean that for any such $v$ all unstable Jacobi fields
along $\gamma_{v}$ would have constant length, which is impossible if
$v\in\mathcal{R}$.

We now know that if $v$ is a generic point for $\mu$, the Birkhoff average of
$a^{u}$ is positive at $v$. It is immediate from this that unstable Jacobi
fields along $\gamma_{v}$ grow exponentially and therefore $\chi^{u}(v)>0$.

A similar argument applies to $\chi^{s}$. ∎

It is clear from their definitions that the exponent $\chi^{u}$ is the
Birkhoff average of the function $-\varphi^{u}$. Finally, even though we do
not use this fact, it is worth noting that $\chi^{s}=-\chi^{u}$ because the
flow $\mathcal{F}$ preserves the contact form on $TT^{1}M$.

##  4\. Proof of Theorem 2.2

The probability measures that are ergodic for $\mathcal{F}$ can be divided
into two classes: those like $\mu_{L}$ for which $\mathcal{R}$ has measure $1$
and $\mathcal{S}$ has measure $0$ and those for which $\mathcal{S}$ has
measure $1$ and $\mathcal{R}$ has measure $0$. The latter class is also
nonempty because $\mathcal{S}$ is compact and invariant.

From Ruelle’s inequality [22] and the fact that $\chi^{u}$ is the only
Lyapunov exponent that can be positive, we know that if $\mu$ is a probability
measure that is ergodic for $\mathcal{F}$, then

$h_{\mu}(\mathcal{F})\leq\int\chi^{u}(v)\,d\mu(v).$

Pesin’s entropy formula [21] tells us that we have equality if $\mu=\mu_{L}$.
We also have equality if $\mu$ is supported on $\mathcal{S}$, since the right
hand side is $0$ in that case. So for any measure $\mu$ supported on the
singular set we have

  * •

$h_{\mu}(\mathcal{F})=0$,

  * •

$\int\varphi^{u}(v)d\mu(v)=0$, and

  * •

$h_{\mu}(\mathcal{F})-\int q\varphi^{u}(v)d\mu(v)=0$ for any $q\in\mathbb{R}$.

Since $\chi^{u}$ is the Birkhoff average of the function $-\varphi^{u}$, we
have

(2) $h_{\mu}(\mathcal{F})+\int\varphi^{u}(v)\,d\mu(v)\leq 0,$

with equality if $\mu=\mu_{L}$ or $\mu$ is supported on the singular set. This
tells us that $\mu_{L}$ and measures supported on $\mathcal{S}$ are
equilibrium states for $q\varphi^{u}$ when $q=1$.

The integral in (2) is negative if $\mu(\mathcal{R})=1$ and $0$ if if
$\mu(\mathcal{S})=1$. If $q>1$, we have

$h_{\mu}(\mathcal{F})+\int q\varphi^{u}(v)\,d\mu(v)\leq 0,$

with equality if and only if $\mu$ is supported on the singular set. This
shows that when $q>1$, the equilibrium states for $q\varphi^{u}$ are precisely
the measures supported on the singular set.

If $\mu=\mu_{L}$, inequality (2) is an equality and the integral on the left
hand side is negative. Hence

$\displaystyle P(q\varphi^{u})$ $\displaystyle\geq
h_{\mu_{L}}(\mathcal{F})+\int q\varphi^{u}(v)\,d\mu_{L}(v)$
$\displaystyle=h_{\mu_{L}}(\mathcal{F})+\int\varphi^{u}(v)\,d\mu_{L}(v)-(1-q)\int\varphi^{u}(v)\,d\mu_{L}(v)>0,$

when $q<1$. If follows that the pressure $P(q\varphi^{u})$ is positive and
therefore anyequilibrium state for $q\varphi^{u}$ must give full measure to
the regular set when $q<1$.

All that remains is to verify that $\mu_{L}$ is the only equilibrium state for
$\varphi^{u}$ that gives full measure to the regular set.

###### Proof of Theorem 2.2.

Let $\mu$ be an ergodic equilibrium state for $\varphi^{u}$ supported on
$\mathcal{R}$. Lemma 3.1 tells us that $\mu$ is hyperbolic. Since
$P(\varphi^{u})=0$ and $\chi^{u}$ is the only exponent that can be positive,
Ruelle’s inequality becomes the equality

$h_{\mu}(\mathcal{F})=-\int_{\mathcal{R}}\varphi^{u}(v)\,d\mu(v).$

Ledrappier and Young prove in Theorem A of [17] that equality holds in the
Ruelle inequality if and only if $\mu$ has absolutely continuous conditional
measures on the Pesin unstable manifolds $W^{u}(v)$, $v\in T^{1}M$. We can
find a set $\Lambda$ with $\mu(\Lambda)>0$ such that if $v\in\Lambda$, then
$v$ is generic for $\mu$ and the Pesin stable and unstable manifolds for $v$
have radius at least $\epsilon$ and depend continuously on $v$. There must be
a Pesin unstable manifold $W$ such that $\lambda_{W}(\Lambda\cap W)>0$, where
$\lambda_{W}$ is the volume measure on $W$ induced by the Sasaki metric on
$T^{1}M$. The set

$G=\bigcup_{v\in\Lambda\cap W}W^{s}(v)$

consists of forward generic points for $\mu$. On the other hand, the
transverse absolute continuity of the Pesin stable manifolds tells us that
$\mu_{L}(G)>0$. Since $\mu_{L}$ is ergodic, $\mu_{L}$-a.e. $v\in G$ is forward
generic for $\mu_{L}$ and therefore forward generic for both $\mu_{L}$ and
$\mu$. Consequently $\mu=\mu_{L}$, since all continuous functions must have
the same integrals with respect to both measures. ∎

### Acknowledgments

This work was carried out in a workshop at the American Institute of
Mathematics. We thank AIM for their support and hospitality.

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