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11institutetext: University of Toronto
11email<EMAIL_ADDRESS>
22institutetext: Adobe Research
22email<EMAIL_ADDRESS>
Project page - https://sayannag.github.io/safari_eccv2024/

#  SafaRi:Adaptive Sequence Transformer for Weakly Supervised Referring
Expression Segmentation

Sayan Nag Work done during internship at Adobe Research.1122 Koustava Goswami
22 Srikrishna Karanam 22

###### Abstract

Referring Expression Segmentation (RES) aims to provide a segmentation mask of
the target object in an image referred to by the text (i.e., referring
expression). Existing methods require large-scale mask annotations. Moreover,
such approaches do not generalize well to unseen/zero-shot scenarios. To
address the aforementioned issues, we propose a weakly-supervised
bootstrapping architecture for RES with several new algorithmic innovations.
To the best of our knowledge, ours is the first approach that considers only a
fraction of both mask and box annotations (shown in Figure 1 and Table 1) for
training. To enable principled training of models in such low-annotation
settings, improve image-text region-level alignment, and further enhance
spatial localization of the target object in the image, we propose Cross-modal
Fusion with Attention Consistency module. For automatic pseudo-labeling of
unlabeled samples, we introduce a novel Mask Validity Filtering routine based
on a spatially aware zero-shot proposal scoring approach. Extensive
experiments show that with just 30% annotations, our model SafaRi achieves
59.31 and 48.26 mIoUs as compared to 58.93 and 48.19 mIoUs obtained by the
fully-supervised SOTA method SeqTR respectively on RefCOCO+@testA and
RefCOCO+testB datasets. SafaRi also outperforms SeqTR by 11.7% (on
RefCOCO+testA) and 19.6% (on RefCOCO+testB) in a fully-supervised setting and
demonstrates strong generalization capabilities in unseen/zero-shot tasks.

###### Keywords:

Weakly-Supervised Referring Expression Segmentation Attention Consistency Mask
Validity Filtering

##  1 Introduction

We consider the problem of referring expression segmentation (RES) [79, 68,
14, 73, 38, 34, 48, 39, 60, 5, 74, 23, 51] where given an image and a text
sentence, the goal is to segment out parts of the image the text is referring
to. Naturally, much existing work in RES blends vision-language understanding
[44, 35, 46, 77, 9, 70, 54, 16] and instance segmentation [40, 6, 13, 20, 2],
allowing for object segmentation using free-form textual expressions rather
than predefined categories [25, 24, 18, 14, 69, 66, 30, 22]. Existing RES
methods are predominantly fully-supervised in nature [14, 73, 68, 24, 18, 22,
14, 60, 38, 5, 79, 39]. Hence, they necessarily require expensive-to-obtain
and extensive manually-annotated segmentation Ground-Truth (GT) masks for
training and typically do not generalize well to unseen/zero-shot scenarios.

As noted above, obtaining large-scale GT masks is very expensive and tedious
which makes it challenging to scale up such methods. To address this, a recent
work (Partial-RES) [53] propose a partially (weakly) supervised solution for
RES task. However, they consider abundant (100%) bounding box annotations to
begin with (Figure 1 and Table 1) and pretrain the model with these boxes on
the Referring Expression Comprehension (REC) task (text referred box
prediction) in a fully-supervised manner. Moreover, they use these 100% box
annotations for filtering pseudo-labels. Their limitations are: (i) in a more
pragmatic and true weakly-supervised setting the percentage of boxes should
equal the percentage of masks, (ii) in their pretraining stage the model is
already aware of the grounding information of the same dataset used in the
weak-supervision stage, (iii) their approach does not account for the cross-
modal region-level interaction between the image and language features which
is crucial for localization tasks [32, 76].

Figure 1: SafaRi achieves state-of-the-art performance in both weakly- and
fully-supervised RES tasks. Although unlike Partial-RES, SafaRi is not
pretrained on fully-supervised REC task, with just 30% annotations, SafaRi
achieves 59.31 mIoU whereas Partial-RES and fully-supervised SeqTR obtains
58.16 and 58.93 mIoUs (see mIoU vs Label-Rate plot). In the weak-supervision
setting, the inclusion of X-FACt (with cross-modal fusion and AMCR components)
and SpARC modules aids SafaRi to demonstrate excellent grounding capabilities
under challenging scenarios where Partial-RES fails (see qualitative
examples). Quantitative results are provided in Tables 2-4.

On the contrary, we investigate a more realistic, challenging and unexplored
problem of Weakly-Supervised Referring Expression Segmentation (WSRES) with
limited human-annotated mask and box annotations, specifically where box %
equals mask % (Figure 1, see Table 1 for comparisons). To tackle our proposed
novel WSRES task, we present SafaRi, an auto-regressive contour-prediction-
based RES method capable of demonstrating excellent performance under
challenging scenarios with few available mask and box annotations. We
specifically choose segmentation as an auto-regressive point prediction task
along the object’s contour instead of an independent pixel prediction
approach. This is because the latter does not account for relationships
between neighbouring pixels and therefore lacks structural information of the
object being segmented, resulting in poor performances in the absence of
abundant mask annotations [53].

In SafaRi, firstly, we introduce a novel Cross-modal (X-) Fusion with Atte-
ntion Consistency (X-FACt) module for facilitating excellent inter-domain
alignment. Since we have only few annotated data, we make the cross-fusion
component in X-FACt to be flexible and parameter efficient. Further, in X-FACt
we actively leverage the cross-attention heatmaps by formulating a
regularization technique that encourages these heatmaps to be consistent with
the referred object in the image. This is enabled by confining the attended
regions within the object contour, enhancing the fidelity of predicted masks.
This is particularly beneficial for the limited annotation scenarios (in
WSRES) where it explicitly helps features improving visual grounding, without
relying on lots of data. Secondly, we systematically devise a novel
bootstrapping strategy (Figure 2) which uses a handful of labeled masks (e.g.,
10%) and iteratively trains the model by utilizing pseudo-masks obtained from
a pseudo-labeling procedure. Finally, in contrast with conventional pseudo-
labeling pipelines which directly use the model predictions as pseudo-labels,
we design a new Mask Validity Filtering (MVF) routine that is responsible for
selection of pseudo-masks (for unannotated data) by validating whether they
spatially align with the boundaries (boxes) of the referred objects or not.
Since we do not possess 100% bounding boxes, we propose SpARC, a novel REC
technique with spatial reasoning capabilities for obtaining these boxes in an
entirely zero-shot manner (Figure 3). Equipped with the above modules, unlike
Partial-RES, our system learns meaningful, transferable and generalizable
representations with rich semantic understanding, empowering it to reason and
make accurate predictions on unseen data (Table 4).

Figure 2: Overview of weakly-supervised bootstrapping setup. It includes an
initial training stage, followed by inference and pseudo-labeling steps.
Filtered pseudo-masks are added to the initial dataset and model is retrained
in an iterative manner.

Method | 100% GT Labels | Fully-supervised | Filtering with
---|---|---|---
Mask? | BBox? | REC Pretraining? | 100% GT BBox?
Partial-RES [53] | ✗ | ✓ | ✓ | ✓
SafaRi (Ours) | ✗ | ✗ | ✗ | ✗

Table 1: Comparison of SafaRi with Partial-RES [53]. Unlike Partial-RES,
SafaRi does not use 100% bounding boxes and considers mask % $=$ box % and it
does not involve fully-supervised REC pretraining. Furthermore, as opposed to
Partial-RES, for the filtering step, SafaRi involves a zero-shot approach to
generate boxes instead of using 100% Ground Truth (GT) boxes. All ✗ evinces
more challenging and realistic WSRES problem setting.

We summarize our main contributions as: (i) To the best of our knowledge, ours
is the first to consider an accurate representation of Weakly-Supervised
Referring Expression (WS-RES) task by considering a novel, more practical and
challenging scenario with limited box and mask annotations where box % equals
mask %. (ii) The novel X-FACt module fosters prediction of high quality masks
by improving cross-modal alignment quality, especially where abundant ground-
truth annotations are not present. (iii) Utilizing SpARC, a novel zero-shot
REC technique, the mask validity filtering stage together with the
bootstrapping pipeline improve system’s self-labeling capabilities. (iv)
Extensive experiments demonstrate the efficacy of SafaRi as it significantly
outperforms baseline models on RES benchmarks. SafaRi also demonstrates strong
generalization capabilities when evaluated on an unseen referring video object
segmentation task in a zero-shot manner. Notably, with just 30% mask-annotated
data, SafaRi achieves 59.31 and 55.83 mIoUs versus 58.93 and 55.64 mIoUs
obtained by the fully-supervised SOTA method SeqTR [79] on RefCOCO+@testA and
RefCOCOg@test sets (Table 2 and Figure 1).

##  2 Related Work

Fully-supervised RES. These methods chiefly focus on enhancing the quality of
multi-modal fusion of extracted vision-language features [22, 38]. Recent
efforts have been made to develop powerful multi-modal transformer-based
methods [15, 66, 33, 30, 39, 79] showing significant improvements over
previous baselines.
Weakly Supervised RES. Weakly supervised instance segmentation methods have
also been investigated in the recent past [1, 17, 78]. However, in the context
of RES, there is one recent study (Partial-RES) which adopts partial or weak-
supervision [53]. They specifically showed that employing a unified multi-
modal transformer model, e.g., SeqTR [79] is more beneficial (as compared to
MDETR [28]) in the presence of limited annotations. In their weakly supervised
RES setting, Partial-RES [53] considered abundant bounding box annotations
albeit limited mask annotations. They utilized a fully-supervised REC
pretrained model to train on limited mask-annotations (Table 1). Furthermore,
for filtering pseudo-masks, they again used the fully available (100%)
bounding box annotations. However, since this involved 100% box annotations,
our proposed setup is much more challenging and truly weakly supervised since
we consider _both box and mask annotations to be limited._ In their pseudo-
labeling step on the data without masks (but with boxes), the model has
already seen the boxes in the REC pretraining stage and thus the grounding
information is already present. Unsupervised Referring Expression
Comprehension (REC). Referring Expression Comprehension (REC) is a grounding
task which involves localizing an object present in an image with respect to a
textual description of that object [28, 80, 21, 37, 71, 52]. With the
introduction of Vision-Language Pretrained (VLP) models [54, 31, 9, 12], it
has been possible to develop Unsupervised or Zero-Shot REC (ZS-REC) methods
[72, 62, 61, 26]. CPT [72] colors region proposal boxes and utilizes a
captioning model for predicting the colored proposals that are linked to the
textual expressions. RedCircle [61] employs visual prompting by drawing
circular contours outside the detected object proposals and subsequently
ranking them based on the obtained CLIP scores. However, it majorly lacks the
understanding of the spatial relationships among the detected proposals.
Referring Video Object Segmentation. Referring Video Object Segmentation
(R-VOS) is a cross-modal task with the objective to segment the target object
in all video frames, referred by a linguistic description. Existing approaches
comprises of (i) bottom-up methods where RES algorithms are applied
independently at the frame-level [73, 59], (ii) top-down methods where a
language grounding model selects the best object tracklet from a candidate set
of tracklets initially constructed by propagating the detected object masks
from key frames [36], and (iii) query-based method which introduce a small set
of object queries that are conditioned on the referring expression for the
target object [67]. However, most of these methods are trained in a fully-
supervised manner on the benchmark R-VOS datasets. Conversely, in this work we
consider a Zero Shot R-VOS (ZS-R-VOS) task where we apply our model (trained
on RES task) in a zero-shot manner.
Sequence-to-Sequence (seq2seq) Modeling in Vision Tasks. Seq2seq modeling has
displayed immense success in Natural Language Processing (NLP) tasks [63, 11,
55, 64, 4, 56]. Taking inspiration from such studies, recent progress have
been made to model vision tasks in a seq2seq manner [7, 8, 70, 65, 45, 39,
79]. One of the foremost studies in this domain is Pix2Seq [7] that proposes
object detection as a seq2seq modeling task conditioned on the observed pixel
inputs. Pix2Seqv2 [8] is an extension of Pix2Seq with an inclusion of instance
segmentation and captioning tasks unified into a single shared interface.
UniTAB [70] employs a unified seq2seq learning framework which is able to
jointly output open-ended text and box representations, facilitating alignment
between words and boxes. Obj2Seq [10] takes objects as inputs and outputs
human pose estimation into sequence-generated form. Finally, SeqTR [79]
proposes to combine visual grounding tasks under a unified framework. Taking
inspiration from these studies, we implement RES with a contour prediction
approach, however, in a weakly-supervised setting.

##  3 SafaRi

We present SafaRi, an adaptive multi-modal sequence transformer with our novel
components: (i) Cross-modal (X-) Fusion with Attention Consistency (X-FACt)
module, (ii) bootstrapped Weak-Supervision with $\gamma$-Scheduling (WSGS),
and (iii) Mask Validity Filtering (MVF) with SpARC.

###  3.1 X-FACt.

X-FACt consists of Fused Feature Extractors (with cross-modal fusion) and
Attention Mask Consistency Regularization (AMCR) as discussed below:
3.1.1 Fused Feature Extractors. We use Swin transformer [42] and RoBERTa [41]
as our image and text feature extractors. A simple concatenation-based
encoding strategy fails to capture cross-modal interactions, leading to poor
fine-grained multi-modal representations. To address this, previous Vision-
Language studies employed cross-modal fusion modules [16, 31, 32] in their
respective frameworks. However, such strategies introduce extra trainable
fusion-specific layers. For instance, the early fusion scheme adopted in GLIP
[32], used 6 vision-language fusion layers and 6 additional BERT layers for
feature enhancement. However, such a scheme is not suitable in the presence of
limited annotations. Therefore, as shown in Figure 3, we introduce a simple
and lightweight fusion mechanism through normalized gated cross-attention into
the layers of uni-modal feature extractors to learn high-quality language
aware visual representations:

Figure 3: Architectural components of SafaRi. (i) We introduce X-FACt,
composed of normalized gated cross-attention based Fused Feature Extractors
and Attention Consistency Mask Regularization (AMCR) for enhancing cross-modal
synergy and spatial localization of target objects. The fused output is
subsequently fed to Sequence Transformer for prediction of contour points.(ii)
We design Mask Validity Filtering (MVF) strategy for choosing valid pseudo-
masks using SpARC module which is a Zero-Shot REC approach with spatial
reasoning capabilities.

$\displaystyle
x_{k}=x_{k}+(1-\beta)\cdot\text{S-MHA}(x_{k-1})+\beta\cdot\text{C-MHA}(\text{S-MHA}(x_{k-1}),y_{k})$
(1) $\displaystyle x_{k}=x_{k}+\text{FFN}(x_{k})$

where $x_{k-1}$ is the output from the $(k-1)^{th}$ layer, S-MHA and C-MHA
represent Self- and Cross-Multi-Head Attention, FFN denotes cross-feed-forward
network, and $\beta$ is a learnable weighted gating parameter initialized from
0. The fused feature then goes into a sequence transformer for decoding the
object contour points as outputs. It is worth noting that unlike previous
works [16, 31] our gated methodology preserves the uni-modal embeddings (i.e.,
self-attention features), ensuring to learn the weighted representations
during training. It also ensures to map linguistic semantic features to the
localized parts of the images, making the system capable of understanding
fine-grained visual features of the objects.

Figure 4: Qualitative differences between cross-attention maps and predicted
masks in the presence and absence of AMCR. Without AMCR, some regions outside
the object boundary are attended which affects the quality of predicted masks.

3.1.2 Attention Mask Consistency Regularization. Feature fusion through cross-
attention strategy is highly effective in attending those parts of the image
which the texts are referring to. However, in Figure 4, we observe that the
attended regions are scattered across the objects including some background
pixels (more pronounced in WSRES task). The lack of fidelity in being able to
attend fine-grained information may be attributed to the lack of training data
(in WSRES). It has also been found in [57] that vision transformers tend to
demonstrate more uniform representations across all layers, enabled by early
aggregation of global information and propagation of features from lower to
higher layers. Moreover, in the absence of abundant masks (and boxes), the
localization capability of the model needs to be further enhanced, especially,
to yield superior pseudo-masks for the bootstrapping stage. Therefore, to
address such issues, we formulate Attention Mask Consistency Regularization
(AMCR) Loss to localize fine-grained cross-attention within the target object:

$\displaystyle\mathcal{L}_{AMCR}=\sum_{b=1}^{{N}}\underbrace{\left(1-\frac{\sum_{i,j}\mathcal{A}_{i,j}\mathcal{M}_{i,j}}{\sum_{i,j}\mathcal{A}_{i,j}}\right)}_{\text{Localization}}+\underbrace{\psi\text{KL}\left(\textbf{U}_{N}||\textbf{Q}_{{N}}\right)}_{\text{Collapse-
Reduction}}$ (2)
$\displaystyle\textbf{Q}_{{N}}=\left\\{\left(\frac{n(\mathcal{A}_{i,j})}{\sum_{i,j}\mathcal{M}_{i,j}}\right)_{b}\forall\;b\in
N\right\\}$

where $\mathcal{A}_{i,j}$ and $\mathcal{M}_{i,j}$ represent the Cross-
Attention map from the last layer and Mask, respectively, $(i,j)$ represents
pixel location, KL denotes KL divergence loss, $\textbf{U}_{{N}}(0,1)$ denotes
uniform distribution of minimum value 0 and maximum value 1,
$\textbf{Q}_{{N}}$ represents computed normalized frequency distribution over
batch with a size $N$. $\psi$ is the loss-balancing term which we empirically
set to 0.001. $\mathcal{L}_{AMCR}$ consists of two terms: a Localization term
and a Collapse-Reduction term. The localization term guarantees accurate
localization and alignment of the generated cross-attention map within the
mask of the object. The collapse-reduction term prevents collapsing of the
cross-attention map within the mask. $\mathcal{L}_{AMCR}$ gets added to the
Cross-Entropy (CE) loss and the total loss becomes:

$\mathcal{L}_{total}=\mathcal{L}_{CE}+\lambda\cdot\mathcal{L}_{AMCR}$ (3)

where $\lambda$ is responsible for weighting the $\mathcal{L}_{AMCR}$ loss
term. We provide more details regarding the formulation of
$\mathcal{L}_{AMCR}$ in the Supplementary.

> Takeaways: (1) X-FACt’s cross-modal fusion scheme is an effective, flexible
> and parameter efficient strategy (when compared to VLP models [16, 31, 32]
> using extra fusion-specific layers) and is particularly beneficial in the
> context of weakly-supervised RES task (Tables 2,4,4,7). (2) The AMCR
> component in X-FACt fosters prediction of high quality masks by spatially
> constraining the cross-attention map within the target object boundary. This
> improves cross-modal region-level alignment quality, especially in the
> limited annotations scenario. Such gains are not achieved upon using L2 loss
> (Table 7).

###  3.2 Weak-Supervision with $\gamma$-Scheduling.

It is worth reiterating that unlike Partial-RES [53], we _do not have box
annotations for the full dataset_. Therefore, we do not _pretrain_ our model
on any box-prediction tasks such as Referring Expression Comprehension (REC)
unlike Partial-RES. Our goal is to best adapt our architecture to the target
RES task given limited mask annotations. Particularly, in a dataset consisting
of $\mathcal{N}$ training image-referring expression pairs, we consider mask
(and box) annotations for $x\%$ data. These $x\%=\\{10\%,20\%,30\%\\}$ samples
are selected randomly from the original training set. Our overall pipeline
(Figure 2) is as follows:
$\bullet$ Step 1: Initial RES Training. In this initial step, we train SafaRi
using using Equation 3 with $x\%$ labeled mask data on the RES task.
Subsequently, we obtain a trained _pseudo-labeler_ with updated model
parameters.
$\bullet$ Step 2: Pseudo-labeling. We infer masks by running inference on the
remaining unlabeled $(100-x)\%$ training samples using model trained from Step
1. Inferred masks are subsequently passed through the proposed Mask Validity
Filtering (MVF) (Section 3.3) to verify the validity of these generated masks
in a zero-shot fashion. Contour points are next sampled from the valid masks
and added to the corresponding image-text pairs as _pseudo-masks_.
$\bullet$ Step 3: Retraining with $\gamma$-Scheduling. We retrain SafaRi
(initialized from previous training) with the updated training dataset
containing both the $x\%$ Ground Truth (GT) Masks ($M$) and Pseudo-Masks
($\hat{M}$) and minimize the final loss $\mathcal{L}_{\textsc{SafaRi}}$ using
a Pseudo-Mask Loss weighting hyperparameter $\gamma$:

$\displaystyle\mathcal{L}_{\textsc{SafaRi}}=\underbrace{\mathcal{L}_{total}^{GT}}_{\text{Loss
with GT
Masks}}+\gamma\cdot\underbrace{\mathcal{L}_{total}^{Pseudo}}_{\text{Loss with
Pseduo-Masks}}$ (4)

The role of $\gamma$ is to balance the amount of GT masks against that of
pseudo-masks during the retraining step. We introduce a $\gamma$-scheduling
strategy to systematically change the value of $\gamma$ over a set of
iterations with the starting (minimum) value of $\gamma$ being
$\gamma_{0}=0.9$ (found empirically through ablation studies) and maximum
value being 1.0. The intuition is to provide more weighting to pseudo-masks as
the model becomes more confident of the predictions as the training steps
progress. We outline the detailed steps in Algorithm 1 in Supplementary.

> Takeaway: Weak-Supervision with $\gamma$-scheduling pipeline boosts self-
> labeling capabilities in the system whereby filtered inferred masks are
> iteratively re-utilized as pseudo-labels in the re-training phase, improving
> mIoU (Tables 2-4).

###  3.3 Mask Validity Filtering with SpARC.

Mask Validity Filtering (MVF) is the process of identifying which predicted
(inferred) masks actually localize the object (present in the image) being
referred to in the text (referring expression) [53]. Partial-RES [53] achieves
this task by using ground-truth bounding-box annotations. However, as noted in
Section 1, our proposed setup considers a true _weakly-supervised_ scenario
where the number of bounding box and mask annotations are equal. Therefore, to
address this gap in the literature, we propose a zero-shot approach to obtain
these boxes to be used in the MVF stage. This can also be formulated as a
Zero-Shot Referring Expression Comprehension (ZS-REC) task. Our proposed MVF
approach is composed of two parts - (i) obtaining bounding boxes using ZS-REC,
and (ii) validating the inferred masks using the obtained bounding boxes.
3.3.1 ZS-REC with SpARC module. We introduce Spatially Aware Red-Box Clip
(SpARC), a ZS-REC module, comprising two components: a proposal scoring
approach with visual prompts (red-box), and a spatial reasoning component
accounting for spatial understanding of proposals in a rule-based manner.
Proposal scoring with red-box prompting: Our ZS framework considers an input
image and the referring expression along with $\mathcal{N}_{\mathrm{B}}$ box
proposals as generated by a pretrained object detector [58]. ZS-REC task can
be conveniently converted to a ZS-retrieval task using an ensemble of image
(visual) prompts with a single text as described below. Each of these visual
prompts (images) must correspond to a single isolated proposal from the
original image generating $\mathcal{N}_{\mathrm{B}}$ such images (visual
prompts). Each proposal object region is isolated by employing a Gaussian blur
on the background and by adding a red box (red border) of uniform thickness
surrounding the object proposal. These image prompts along with the referring
expression is passed through CLIP to obtain the CLIP score. Notably, using a
red border acts as a positive visual prompt highlighting the target area,
whereas blurring the background reduces the impact of weakly related
information. However, these CLIP scores do not consider the spatial
understanding of the objects in the different bounding boxes. Therefore,
injecting spatial understanding is important which we discuss next.
Spatial Reasoning component: A major limitation of CLIP for the ZS-REC task is
the lack of fine-grained spatial understanding. In Figure 3, the text "_black
dog_ " refers to the two black dogs in the picture. CLIP will generate high
scores for both these proposal regions since both fall under the class "_black
dog_ ". Furthermore, although addition of visual prompt improves detection
performance [61], it neither incorporates nor enhances spatial understanding
capabilities of the model. To mitigate this, we introduce a simple rule-based
approach. We divide the referring expression into two constituents - noun
chunks and their relations. In the previous example, it refers to the "_black
dog_ " in the picture which is the subject. The phrases connecting these noun
chunks form the relationships (e.g., _right/east_ , _left/west_ ,
_smaller/tinier/further_ , _bigger/larger/closer_ , _between_ ,
_within/inside_ , _above/north/top_ , _below/under/south_ , _back/behind_ ,
and _front_). In the presence of two boxes referring to the _black dog_ , the
probability of the proposal on the left will be the most (as decided based on
the location of the centroid of the box computed using the obtained box
coordinates). To provide more clarity on this, we consider the example:
"_black dog on the left of a white dog_ ", relation $R$ between two nouns
black dog ($X$) and white dog ($Y$) is "_left_ ". Coordinates of proposals $X$
and $Y$, and relation $R$ are the inputs of Spatial Reasoning component. If
center point of box $X$ is to the "_left_ " of that of box $Y$, the output
probability $\text{Pr}[R(X,Y)]=1$, otherwise 0. These probabilities are
multiplied with the CLIP scores of the respective proposals. Note that in rare
occasions where the referring texts in the RefCOCO datasets do not have any
spatial relations, we resort to using only CLIP scores for scoring the
proposals. More examples, with those of complex relations are given in
Supplementary.
3.3.2 Validation of Inferred Masks with SpARC. In the Pseudo-labeling step,
SafaRi predicts contour points which are connected to generate binary masks.
We generate a bounding box from the outermost (top-most, bottom-most, right-
most, left-most) points of each mask and compute Dice Similarity Coefficient
(DSC) [49] between the generated box and the box obtained using the ZS-REC
step using SpARC. We reject the noisy pseudo-masks with DSC value less than
$\tau=0.1$ (ablation in Supplementary). Contour points are resampled from the
filtered pseudo-masks and added to the training set (Figure 2).

> Takeaways: (1) Visual prompting with red border together with the spatial
> awareness component in SpARC improves mIoU (as demonstrated in Table 5). (2)
> Mask Validity Filtering with SpARC enhances the quality of training samples
> in the retraining stage, which has a positive effect on the model
> performance (Tables 2-4) for the weakly-supervised RES task.

##  4 Experiments and Results

###  4.1 Datasets, Implementation Details and Metrics

4.1.1 Dataset. For RES, we conduct our experiments on the three major RES
datasets: RefCOCO [75], RefCOCO+ [75], and RefCOCOg [50, 47]. For Zero-shot
Transfer to Referring Video Object Segmentation (ZS-R-VOS), we conduct zero-
shot (ZS) evaluations on two popular R-VOS datasets: RefDAVIS17 [29] and
JHMDB-Sentences [19].
4.1.2 Implementation. We use AdamW optimizer [43] with a batch size of 128. We
use an initial learning rate (LR) of 5e-4 with
$(\beta_{1},\beta_{2})=(0.9,0.98),\epsilon=10^{-9}$, and Multi-Step LR Warmup
of 5 epochs, decay steps of 75, and a decay ratio of 0.1. We use A100 40GB
GPUs for all experiments.
4.1.3 Metrics. For RES, we report mean Intersection-over-Union (mIoU) values.
For ZS-R-VOS, we report mean of region similarity ($\mathcal{J}$) and contour
accuracy ($\mathcal{F}$) denoted as $\mathcal{J}\&\mathcal{F}$ on RefDAVIS17,
and mIoU scores on JHMDB-Sentences. Additional dataset and implementation
details are provided in Supplementary.

Label-Rate | Method | Vis. Backbone | RefCOCO | RefCOCO+ | RefCOCOg
---|---|---|---|---|---
Mask | BBox | val | testA | testB | val | testA | testB | val-g | val-u | test-u
Fully Supervised Models
100% |  |  LTS [27] | DN53 | 65.43 | 67.76 | 63.08 | 54.21 | 58.32 | 48.02 | - | 54.40 | 54.25
|  VLT [14] | DN56 | 65.65 | 68.29 | 62.73 | 55.50 | 59.20 | 49.36 | 49.76 | 52.99 | 56.65
|  ResTR [30] | ViT-B | 67.22 | 69.30 | 64.45 | 55.78 | 60.44 | 48.27 | 54.48 | - | -
|  SeqTR [79] | DN53 | 67.26 | 69.79 | 64.12 | 54.14 | 58.93 | 48.19 | - | 55.67 | 55.64
| SafaRi (Ours) | Swin-B | 73.35 | 75.02 | 70.71 | 63.03 | 65.81 | 57.64 | - | 62.42 | 62.74
| $\Delta_{\text{Ours - SeqTR}}$ | - |  6.09 $\uparrow$ |  5.23 $\uparrow$ |  6.59 $\uparrow$ |  8.89 $\uparrow$ |  6.88 $\uparrow$ |  9.45 $\uparrow$ | - |  6.75 $\uparrow$ |  7.10 $\uparrow$
3-13 |  |  PolyFormer [73]† | Swin-B | 75.96 | 77.09 | 73.22 | 70.65 | 74.51 | 64.64 | - | 69.36 | 69.88
|  | SafaRi† (Ours) | Swin-B | 77.21 | 77.83 | 75.72 | 70.78 | 74.53 | 64.88 | - | 70.48 | 71.06
|  | $\Delta_{\text{Ours - PolyFormer}^{{\dagger}}}$ | - |  1.25 $\uparrow$ |  0.74 $\uparrow$ |  2.50 $\uparrow$ |  0.13 $\uparrow$ |  0.02 $\uparrow$ |  0.24 $\uparrow$ | - |  1.12 $\uparrow$ |  1.18 $\uparrow$
Weakly Supervised Models
30% | 100% |  Partial-RES [53]$\spadesuit$ | Swin-B | 66.24 | 68.39 | 63.57 | 54.37 | 58.16 | 47.92 | - | 54.69 | 54.81
30% | SafaRi (Ours) | Swin-B | 67.04 | 69.17 | 64.23 | 54.98 | 59.31 | 48.26 | - | 55.72 | 55.83
| $\Delta_{\text{Ours - Partial-RES}}\spadesuit$ | - |  0.80 $\uparrow$ |  0.78 $\uparrow$ |  0.66 $\uparrow$ |  0.61 $\uparrow$ |  1.15 $\uparrow$ |  0.34 $\uparrow$ | - |  1.03 $\uparrow$ |  1.02 $\uparrow$
20% | 100% |  Partial-RES [53]$\spadesuit$ | Swin-B | 65.20 | 67.43 | 62.85 | 53.78 | 57.52 | 47.39 | - | 53.94 | 54.02
20% | SafaRi (Ours) | Swin-B | 65.88 | 67.96 | 63.24 | 54.23 | 58.07 | 47.67 | - | 54.45 | 54.61
| $\Delta_{\text{Ours - Partial-RES}}\spadesuit$ | - |  0.68 $\uparrow$ |  0.53 $\uparrow$ |  0.39 $\uparrow$ |  0.45 $\uparrow$ |  0.55 $\uparrow$ |  0.28 $\uparrow$ | - |  0.51 $\uparrow$ |  0.59 $\uparrow$
10% | 100% |  Partial-RES [53]$\spadesuit$ | Swin-B | 64.01 | 65.89 | 61.68 | 52.85 | 56.01 | 46.27 | - | 52.73 | 52.68
10% | SafaRi (Ours) | Swin-B | 64.02 | 65.91 | 61.76 | 52.98 | 56.24 | 46.48 | - | 52.91 | 52.94
| $\Delta_{\text{Ours - Partial-RES}}\spadesuit$ | - |  0.01 $\uparrow$ |  0.02 $\uparrow$ |  0.08 $\uparrow$ |  0.13 $\uparrow$ |  0.23 $\uparrow$ |  0.21 $\uparrow$ | - |  0.18 $\uparrow$ |  0.26 $\uparrow$

Table 2: Comparison with the state-of-the-arts on the RES task. SafaRi
substantially outperforms SOTA SeqTR [79] in the fully-supervised benchmark.
SafaRi also yields significant gains over baseline Partial-RES [53] even
without using 100% box annotations in the WSRES task. ${\dagger}$ means
trained on extra data combining RefCOCO datasets [39]. $\spadesuit$ indicates
our reimplementation of Partial-RES with Swin-B backbone where we get better
mIoUs than their reported values [53]. Differences shown in blue.

###  4.2 Main Results

4.2.1 Referring Expression Segmentation (RES). We compare SafaRi with the
state-of-the-art methods on the RefCOCO/+/g validation and test sets and
report the mean IoU (mIoU) in the Table 2 under different label-rates.
Full Supervision: SafaRi outperforms existing benchmark methods on each split
of the three datasets by a significant margin (Table 2). For a fair comparison
with PolyFormer [39], we additionally train SafaRi on a combined dataset after
removing all data from validation and test sets [39]. Unlike PolyFormer,
SafaRi is not pretrained on the REC task. However, SafaRi still outperforms
PolyFormer on all three datasets. Note that we report fully-supervised results
to show an upper-bound on the performance of the weakly-supervised models.
Weak Supervision: We consider label rates of 10%, 20%, and 30% for the WSRES
task (Table 2). Despite methods such as Partial-RES using 100% box
annotations, SafaRi shows substantial improvements over Partial-RES (note that
Partial-RES originally used DN-53 backbone and we re-implemented with Swin-B
backbone for fair comparisons) for all three label-rates. With just 30%
labeled data and without any REC pretraining, SafaRi often surpasses the
fully-supervised performance of SeqTR on RefCOCO/+g validation sets (e.g.,
59.31 vs 58.93 on RefCOCO+@testA, 55.83 vs 55.64 on RefCOCOg@test sets).
Weakly-supervised SafaRi significantly outperforms other fully-supervised
baselines (e.g., VLT [14] and LTS [27]). These results bespeak the role of our
proposed X-FACt (containing cross-modal fusion together with AMCR) and SpARC
modules in obtaining state-of-the-art results in the limited annotation
scenarios.
4.2.2 Zero-shot Referring Video Object Segmentation (ZS-R-VOS). We conduct
evaluation on the R-VOS datasets (RefDAVIS17 and JHMDB) as a ZS transfer task.
We consider the video frames as a sequence of images without involving any
temporal information while predicting masks. As shown in Table 4, SafaRi
achieves state-of-the-art results with just the spatial information,
displaying strong generalization capabilities.

We also conduct evaluations on the ZS-REC task with our proposed SpARC module
which significantly outperforms SOTA RedCircle [61] achieving 39.3% and 16.2%
gains on RefCOCO@testB and RefCOCO+@testB sets respectively. Results and
visualizations are given in Supplementary.

Method | Vis. Backbone | Eval. | RefDAVIS17 | JHMDB
---|---|---|---|---
$\mathcal{J}\&\mathcal{F}$ | mIoU
Fully Supervised Models (Label-Rate = 100%)
ReferFormer [67] | Swin-L | FT | 60.5 | -
MTTR [3] | Vid-Swin-T | FT | - | 36.6
ReferFormer [67] | Vid-Swin-B | FT | 61.1 | 43.7
PolyFormer [39] | Swin-B | ZS | 60.9 | 42.4
SeqTR [79] | DN-53 | ZS | 53.5 | 34.9
SafaRi-100 (Ours) | Swin-B | ZS | 61.3 | 43.2
$\Delta_{\text{Ours - SeqTR}}$ | - | ZS |  7.8 $\uparrow$ | 8.3$\uparrow$
Weakly Supervised Models
Partial-RES-30 [53] | Swin-B | ZS | 52.3 | 34.8
SafaRi-30 (Ours) | Swin-B | ZS | 55.3 | 38.1
$\Delta_{\text{Ours - Partial-RES-30}}$ | - | ZS |  3.0 $\uparrow$ | 3.3$\uparrow$
Partial-RES-10 [53] | Swin-B | ZS | 51.5 | 34.3
SafaRi-10 (Ours) | Swin-B | ZS | 53.1 | 36.4
$\Delta_{\text{Ours - Partial-RES-10}}$ | - | ZS |  1.6 $\uparrow$ | 2.1$\uparrow$

Table 3: Evaluation performance of SafaRi on RefDAVIS17 and JHMDB validation
datasets. SafaRi achieves significant gains in zero-shot settings for both
datasets. FT and ZS indicate Fine-Tuned and Zero-Shot, respectively.
Differences in blue.

Label-Rate | Cross-Attn. | AMCR | RefCOCO
---|---|---|---
val | testA | testB
100 | ✗ | ✗ | 68.84 | 70.59 | 65.85
✓ | ✗ | 72.41 | 74.65 | 69.96
✓ | ✓ | 73.35 | 75.02 | 70.71
30 | ✗ | ✗ | 60.91 | 64.06 | 57.73
✓ | ✗ | 64.72 | 67.55 | 60.91
✓ | ✓ | 67.04 | 69.17 | 64.23
20 | ✗ | ✗ | 58.84 | 61.79 | 57.02
✓ | ✗ | 61.25 | 64.68 | 59.47
✓ | ✓ | 65.88 | 67.96 | 63.24
10 | ✗ | ✗ | 54.06 | 55.64 | 52.61
✓ | ✗ | 60.11 | 62.18 | 57.95
✓ | ✓ | 64.02 | 65.91 | 61.76

Table 4: Impact of Cross-Attention and AMCR (in the X-FACt module) on SafaRi’s
performance evaluated on RefCOCO val, testA, and testB sets for 100%, 30%,
20%, and 10% label-rates. Best results are highlighted.

###  4.3 Ablation Study

4.3.1 Impact of X-FACt. We show the impact of X-FACt containing Cross-
Attention (CA) based fusion and the AMCR components in Table 4. Inclusion of
CA is shown to improve mIoU values consistently across varying label-rates.
The impact of AMCR is more pronounced in the cases of limited annotations. For
instance, with a label-rate of 100%, addition of AMCR boosts mIoU by 0.94
points on RefCOCO@val, whereas with a label-rate of 10%, this difference
increases substantially to 3.91. Additionally, in Figure 4 we demonstrate that
incorporating AMCR qualitatively improves both the cross-attention maps and
the predicted masks, highlighting the efficacy of AMCR in our pipeline. We
further assess the importance of AMCR loss balancing factor ($\lambda$) in the
Figure 5(b) when evaluated on RefCOCO@val. Increasing $\lambda$ improves mIoU
initially and the maximum is achieved at 0.4, beyond which the performance
drops significantly. Moreover, in Table 7 we show that despite being
lightweight, our proposed fusion scheme in X-FACt is superior in performance
as compared to (conventional) ALBEF-like fusion which contains extra fusion-
specific layers.

$1$$3$$5$$7$$35$$45$$55$$65$runmIoU$\gamma_{0}=0.9$$\gamma_{0}=0.8$

(a) mIoU vs runs.

$0.2$$0.4$$0.6$$35$$45$$55$$65$$\lambda$mIoUSafaRi-30SafaRi-10

(b) mIoU vs $\lambda$.

Figure 5: Impact of $\gamma$-scheduling under different initial values
($\gamma_{0}$) and AMCR balancing factor ($\lambda$) when evaluated on
RefCOCO@val at 30% and 10% mask labels.

Red-box | Spatial | Blur | Crop | Ref@val | Ref+@val | Refg@val
---|---|---|---|---|---|---
✗ | ✗ | ✗ | ✓ | 63.11 | 47.48 | 49.43
✗ | ✗ | ✓ | ✗ | 63.78 | 49.66 | 51.48
✓ | ✗ | ✗ | ✗ | 64.35 | 51.16 | 52.87
✓ | ✗ | ✓ | ✗ | 65.29 | 52.51 | 54.22
✗ | ✓ | ✗ | ✓ | 64.12 | 50.19 | 51.80
✗ | ✓ | ✓ | ✗ | 65.94 | 52.83 | 54.13
✓ | ✓ | ✓ | ✗ | 67.04 | 54.98 | 55.72

Table 5: mIoUs with various prompts for SpARC at 30% label-rate. Best results
are obtained when both red-box and blurring are used with the spatial
reasoning component.

Label-Rate | L2 | AMCR | Ref@val | Ref+@val | Refg@val
---|---|---|---|---|---
30 | ✓ | $-$ | 63.48 | 51.31 | 52.16
$-$ | ✓ | 67.04 | 54.98 | 55.72
10 | ✓ | $-$ | 59.23 | 48.29 | 48.83
$-$ | ✓ | 64.02 | 52.98 | 52.91

Table 6: Comparison of proposed AMCR with L2 loss for 10% and 30% annotations.

Fusion | Label- | RefCOCOg |  $\Delta$ parameter
---|---|---|---
Scheme | Rate | val | increase (%) $\downarrow$
X-FACt | 30 | 55.72 | 1.82
(Ours) | 100 | 62.42
ALBEF- | 30 | 54.29 | 29.65
like | 100 | 60.17

Table 7: X-FACt vs ALBEF-like fusion for 30% and 100% annotations.

4.3.2 Retraining with MVF using SpARC. In the presence of limited mask (and
box) annotations, retraining with $\gamma$ scheduling leads to a marked
improvement in the mIoU values as depicted in Figure 5(a). Here, we consider
label-rates of 30% (solid lines) and 10% (dashed lines) with two different
$\gamma_{0}$ values denoting the starting values of the $\gamma$ variable. We
note that as the number of retraining steps (runs) increases, the mIoU values
(for both $\gamma_{0}$ values) in each label-rate increases. However, with
$\gamma_{0}=0.9$ we achieve slightly better performance. Additionally, we
evaluate the effectiveness of the MVF stage with SpARC in Supplementary.
Results indicate that without MVF, generated pseudo-labels cannot be validated
which negatively impacts model’s performance. In contrast, when pseudo-labels
are validated using MVF, the mIoU values on RefCOCO@val set are shown to
improve by 4.36 mIoU, demonstrating the importance of MVF.
4.3.3 Impact of various components of SpARC. In Table 5 we assess the impact
of individual components of SpARC. We note that red-box prompting and blurring
together boost performance. Adding spatial awareness via the rule-based
reasoning module also substantially improves mIoU on the WSRES task.
4.3.4 AMCR vs L2 loss in X-FACt. To evaluate the impact of the AMCR component
in X-FACt, we analyze SafaRi’s performance after replacing the proposed
formulation of AMCR with a L2 loss. As shown in Table 7, we observe that using
L2 (in place of AMCR) leads to substantially poor performance, as compared to
when AMCR is used in X-FACt. This corroborates the strong localization
capabilities introduced in the system by the AMCR formulation which is not
achieved with a L2 loss.
4.3.5 Impact of Label Rates. We notice that as the label-rate increases from
10% to 100%, the performance of the model improves significantly across all
the datasets (see Table 2 and Figure 1). Remarkably, with just 30% annotated
data, SafaRi achieves 59.31 mIoU versus 58.93 mIoU obtained by the fully-
supervised SeqTR on RefCOCO+@testA set.

We provide additional ablations on the impacts of MVF, gated cross-attention
parameter, and DSC thresholding parameter in Supplementary.

Figure 6: Cross-attention Maps and corresponding predictions showing strong
cross-modal alignment learned by SafaRi.

Figure 7: Predictions with varying label-rates. With increasing mask
annotations %, prediction quality improves.

###  4.4 Qualitative Assessment and Error Analysis

4.4.1 Backbone Cross-Attentions and Predicted Masks. In Figure 7, we show how
cross-modal attention maps attend to different objects in the images, guided
by the referring texts. The cross-attention scores between the image and the
associated expression are extracted and bilinearly interpolated to match the
image dimension and superimposed on the original image. Notably, strong cross-
modal interactions aid in predicting high-quality masks as displayed in Figure
7. In Figure 9, we also provide high-quality segmentation masks generated by
SafaRi for the ZS-R-VOS tasks without any finetuning on the respective
datasets.
4.4.2 Varying Label-Rates. In Figure 7, we display segmentation masks under
varying label-rates using SafaRi. We clearly notice that the quality of masks
improve substantially with an increase in the ground-truth annotations.
4.4.3 Varying Retraining Steps. In Figure 8 we show that with increasing
retraining steps (runs), SafaRi becomes more confident which results in
significant qualitative improvements in the predictions. For example, in
Figure 8 although SafaRi could not recognize the magazine partially behind
laptop in the first step, it was successful in predicting the mask accurately
in the final step - this shows the efficacy of the retraining stage.
4.4.4 Comparisons with Partial-RES. We provide qualitative examples for 30%
label-rate in Figure 1. It is evident from Figure 1 that SafaRi succeeds in
challenging cases requiring extensive linguistic understanding (attained with
X-FACt and SpARC modules) where Partial-RES [53] fails.

We provide extended versions of these qualitative visualizations in
Supplementary. Additionally, we show examples of cases in Supplementary where
SafaRi occasionally fails to focus on the referred objects. These cases
typically consists of vastly hindered objects in cluttered environments,
especially in low-light blurry conditions and tiny objects.

Figure 8: Examples of masks with increasing WSRES bootstrapping runs (steps)
for 10% annotations. We see significant improvements in localization
capabilities with an increase in retraining steps illustrating the efficacy
our approach. Figure 9: Predicted masks on RefDAVIS17 and JHMDB datasets in a
zero-shot setting with SafaRi-30 (trained with 30% annotations).

##  5 Conclusion

We present a weakly-supervised learning framework for RES considering limited
mask (and box) annotations and employing a contour-based sequence prediction
approach. Unlike Partial-RES [53], we do not consider box annotations to be
fully available and therefore we do not pretrain our model on the fully-
supervised REC task. We incorporate lightweight gated cross-modal attention in
the feature backbones along with an Attention Mask Consistency Regularization
module to facilitate strong cross-modal alignment and improve the quality of
predicted masks. We introduce a bootstrapping pipeline with self-labeling
capabilities where pseudo-labels are validated using our proposed Mask
Validity Filtering approach. We conduct extensive experiments to demonstrate
that SafaRi consistently achieves state-of-the-art performance across all RES
datasets. Finally, we demonstrate excellent generalization capabilities of
SafaRi on zero-shot referring video-object segmentation task. Extending our
approach to multi-image and video settings can be looked upon as a promising
future work.

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Little Conformal Symmetry

Rachel<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>, Kitran Colwell†, and John Terning‡

Department of Physics, University of California, One Shields Ave., Davis, CA
95616

Abstract

> We explore a new class of natural models which ensure the one-loop
> divergences in the Higgs mass are cancelled. The top-partners that cancel
> the top loop are new gauge bosons, and the symmetry relation that ensures
> the cancellation arises at an infrared fixed point. Such a cancellation
> mechanism can, a la Little Higgs models, push the scale of new physics that
> completely solves the hierarchy problem up to 5-10 TeV. When embedded in a
> supersymmetric model, the stop and gaugino masses provide the cutoffs for
> the loops, and the mechanism ensures a cancellation between the stop and
> gaugino mass dependence of the Higgs mass parameter.

##  1 Introduction

The discovery of a $125$ GeV Higgs boson by the CMS and ATLAS experiments
fills in the final particle in the Standard Model (SM). In order to achieve
the measured Higgs mass, however, the SM requires a fine-tuning of the mass
parameter to cancel against divergent loop contributions. Meanwhile, LEP
constraints suggest that higher dimension operators are suppressed by a scale
around 10 TeV [1]. This “little” hierarchy problem provides a compelling
motivation to study physics beyond the SM which can cancel the divergent loop
contributions and make the theory technically natural. SUSY is the prime
example of a natural theory, with stop squark loops canceling the divergent
part of the top loops, and gauginos doing the same for the gauge loops.
However, with the absence (so far) of any kind of colored top-partners at the
LHC there has been a great rush to abandon naturalness as a guiding principle
in searching for extensions of the the SM. This has lead to an advance
(retreat?) into anthropic landscapes, which seem to be highly non-predictive
for particle physics.

Before we abandon naturalness, it would be good to have some idea of how many
classes of natural theories exist that have not yet been excluded by LHC data.
For example, Little Higgs theories provide an alternative class of natural
theories where a global symmetry provides a top-partner which is a fermion
(rather than a scalar as happens in SUSY). There has also been recent progress
finding new types of natural models in the class of “neutral naturalness”
models [2], where a discrete symmetry provides a fermionic top-partner that is
color neutral.

Here we examine yet another type of natural theory where the top-partner is a
gauge boson111A model with a gauge boson top partner has already appeared [3],
but in that case the gauge boson was a superpartner of the top quark.. In
order to cancel the quadratic divergence in the top loop, the top partner
gauge boson needs to have a coupling that is related to the top Yukawa
coupling. In our scenario this relation between the couplings is accidental in
that it is not the result of a symmetry of the Lagrangian, but arises only at
an infrared fixed point. As in [4], we expect that this extra bit of conformal
symmetry can only cancel divergent diagrams at one-loop, since conformal
symmetry does not by itself ensure that scalar masses vanish. With a one-loop
cancellation we can proceed as in Little Higgs models, and push the scale of
new physics that completely solves the hierarchy problem up to 5-10 TeV. For
example, superpartners could have 10 TeV masses, so SUSY would ensure that the
Higgs mass is kept below 10 TeV, and our new mechanism, which we will refer to
as Little Conformal Symmetry, can ensure the cancellations that keep the Higgs
below 1 TeV.

Of course, there have been many attempts to use conformal symmetry to address
the hierarchy problem [5], but these attempts typically stumble on imperfect
cancellations [4, 6], or ultimately on the existence of the Planck scale. The
underlying problem, as we have said, is that while the vanishing of scalar
masses is necessary for the existence of conformal symmetry, conformal
symmetry does not, by itself, enforce a vanishing scalar mass. However if we
only need the cancellation to work at one-loop, then there is still hope that
conformal symmetry can be useful. There is also a further problem that
conformal symmetry cannot help with: what determines the cutoffs in the
divergent loops? This can only be answered in a theory where the cutoffs are
calculable, so we will examine the case where Little Conformal Symmetry
embeded in a supersymmetric model with gauge mediation where the ratio of stop
and gaugino masses is fixed by gauge couplings.

This paper is organized as follows: in the next section we describe a simple
toy model the illustrates the mechanism, then we turn to a more realistic
model which does not assume that the different loop cutoffs are equal. Finally
we present our conclusions and make suggestions for searching for a completely
realistic model.

##  2 A Toy Model

The contributions to the quadratic divergence of the Higgs mass come from the
following diagrams shown in Fig. 1.

$-im_{H}^{2}(p^{2})=$$+\quad$$+\quad$

Figure 1: Quadratically divergent diagrams that contribute to the Higgs mass

Including only the leading fermionic contribution from the top quark with
$N_{c}$ colors, the quadratic correction to the Higgs mass is [7]:

$\displaystyle-
im_{H}^{2}(0)=\left[6\lambda-2N_{c}\,y_{t}^{2}+3g_{i}^{2}C_{2}^{i}(H)\right]\int^{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}}$
(2.1)

where $\lambda$ is the Higgs quartic coupling, $y_{t}$ is the top Yukawa, the
$i$ index runs over all the gauge couplings of the Higgs, and $C_{2}^{i}(H)$
is the quadratic Casimir of the Higgs field representation in the
$i^{\text{th}}$ gauge group [8].

Long ago, Veltman [9] suggested that there could be a cancellation of these
disparate contributions. However, no symmetry was found that would ensure the
cancelation, and the required top mass was the (then) almost unimaginably
large value of 69 GeV. A further problem with Veltman’s proposal is that if
some new physics cuts off the integrals, then there is no guarantee that the
cutoffs of the three different types of loops would be the same. In the
context of SUSY, the top loop is cut off by the stop squark mass, while the
gauge loops are cut off by the gaugino masses, which are typically not equal
in models of dynamical SUSY breaking.

Let us first consider how the top loop could be cancelled by a new non-Abelian
gauge boson loop. For this to occur the Higgs would have to be embedded in a
multiplet of a new gauge group. For the Yukawa coupling to be gauge invariant,
one or both of the left-handed and right-handed tops would also have to
transform under the new gauge symmetry. Let us also assume for now that the
cutoffs of the two loop integrations are the same. (We will return to this
point in the next section, where the cutoffs will be superpartner masses that
will not be equal.) Ensuring the cancellation gives us a requirement that the
top Yukawa coupling, $y_{t}$, is related to the new gauge coupling $g_{N}$.
Surely no symmetry of a Lagrangian could force such a relation, but if the top
Yukawa coupling is at an infrared fixed point that was determined by the value
of the new gauge coupling, which itself is at an infrared fixed point, then
there is a relation between the two couplings.

In order to construct a simple toy model that illustrates this mechanism, we
will for now ignore the effects of the SM gauge groups and the quartic Higgs
coupling. For concreteness let us embed the Higgs and the right-handed top in
a fundamental and anti-fundamental of a $SU(N)$ gauge group. The
$\beta$-function for the top Yukawa coupling in this theory is [10, 11]:

$\frac{dy_{t}}{d\ln\mu}=\frac{1}{16\pi^{2}}\left[\frac{1}{2}\left(2+n_{D}\right)+N_{c}\right]y_{t}^{3}-\frac{3}{16\pi^{2}}\left[C_{2}(t_{R})+C_{2}(t_{L})\right]g_{N}^{2}y_{t}$
(2.2)

where $C_{2}(F)$ is the quadratic Casimir of the representation of fermion $F$
under $SU(N)$, and $n_{D}$ is the number of Higgs doublets, so in the SM
$N_{c}=3$ and $n_{D}=1$ and the first coefficient reduces to the standard
result $9/32\pi^{2}$. In our toy model $C_{2}(t_{L})=0$ and $n_{D}=N$, so a
Yukawa fixed point occurs when [12]

$0=\left(4+\frac{N}{2}\right)y_{t*}^{2}-3\,C_{2}(t_{R})g_{N*}^{2}$ (2.3)

Cancelling the quadratic divergence simultaneously requires

$0=-2N_{c}\,y_{t*}^{2}+3\,C_{2}(H)g_{N*}^{2}~{},$ (2.4)

which gives us a relation between the Casimirs of the Higgs field and right-
handed top quark:

$\frac{C_{2}(H)}{C_{2}(t_{R})}=\frac{12}{8+N}~{},$ (2.5)

and since the $t_{R}$ and the Higgs are in conjugate $SU(N)$ representations
the solution is $N=4$.

For the toy model to be consistent we also need the $SU(4)$ gauge coupling to
be at a fixed point. The two-loop gauge $\beta$-function is

$\frac{dg_{N}}{d\ln\mu}=-\frac{1}{16\pi^{2}}\left(b\,g_{N}^{3}+c\,g_{N}^{5}+d\,g_{N}y_{t}^{2}\right)~{}.$
(2.6)

When the gauge group is asymptotically free, i.e. $b>0$, then often $c<0$, and
if $b$ is small there is a perturbative Banks-Zaks fixed point [13] for
$y_{t}=0$. There is no general theorem determining whether there are fixed
points for $y_{t}\neq 0$, but they can be easily found by scanning over the
possible gauge representations of the matter fields. Generically there are
multiple solutions for fixed points of the coupling $g_{N}$ at the fixed point
$y_{t*}$, depending on the matter content of the gauge theory, we will call
them $g_{N*i}$. In order to make an interesting model we would arrange $SU(4)$
to break at some scale around $\Lambda\sim$ 5-10 TeV; that is, parts of the
extended $t_{R}$ and Higgs multiplets can get masses at this scale, while the
components corresponding to the SM $t_{R}$ and Higgs remain light. The gauge
bosons could have masses somewhere between 1 TeV and $\Lambda$. Given the
measured value of $y_{t}$ we could run it up towards the UV, and at each RG
scale $\mu_{i}$ where $y_{t}(\mu_{i})$ satisfies Eq. (2.3) with
$g_{N*}=g_{N*i}$ we have a possible consistent model.

##  3 A More Realistic Toy Model

If we try to directly apply the mechanism of the previous section to the SM,
we immediately run into a problem: the QCD contribution to top Yukawa $\beta$
function (2.2) is much larger that the Yukawa contribution. This implies that
the new gauge group would not dominate at a fixed point and this would spoil
our cancellation. In order to get around this we can embed color $SU(3)_{c}$
in $SU(N)$, and take both $t_{L}$ and $t_{R}$ to transform under conjugate
representations of $SU(N)$. Going further we can take a semi-simple gauge
group $SU(N)\times SU(N^{\prime})$ with the understanding that $SU(3)_{c}$ is
the diagonal subgroup of $SU(3)_{L}\times SU(3)_{R}\subset SU(N)\times
SU(N^{\prime})$. We will also assume that the top quarks are charged under an
additional global $SU(M)$ symmetry to enhance the otherwise small top loop
contribution to both (2.2) and (2.4).

In order to connect to the pheonomenology of the SM and ensure anomaly
cancellation for the $SU(N)$ and $SU(N^{\prime})$ gauge fields, we must
introduce the spectator fermions $b_{R}$ that contain the right-handed bottom
quark. For now we will only consider Yukawa couplings that give masses to the
top quark. A summary of the charge assignments for the third generation quarks
and Higgs fields consistent with anomaly cancellation are given in Table 1 (a
second choice that yields asymptotic freedom and anomaly cancellation
interchanges the $b_{N}$ and $b_{M}$ representations).

| $SU(N)$ | $SU(N^{\prime})$ | $SU(M)$
---|---|---|---
$\rule{0.0pt}{12.91663pt}\bar{t}_{R}$ | $\overline{\Box}$ | $1$ | $\overline{\Box}$
$H_{u}$ | $\Box$ | $\overline{\Box}$ | $1$
$Q_{L}$ | $1$ | $\Box$ | $\Box$
$\bar{b}_{N}$ | $\overline{\Box}$ | $1$ | $1$
$\bar{b}_{M}$ | $1$ | $\overline{\Box}$ | $1$
Table 1: Top and Higgs charges under a bifundamental Higgs

The one-loop running of a Yukawa coupling can be computed from the diagrams in
Figure 2.

$\displaystyle\frac{dy_{t}}{d\ln\mu}=$ $\quad+\quad$ $\quad+\quad$
$\quad+\quad$ $\quad+\quad$ $\quad+\quad$
$\quad+\quad$ $\quad+\quad\text{$\bar{t}_{R}$ leg corrections}$

Figure 2: $\beta$ function for top Yukawa coupling

The index structure of our theory causes the 1PI contribution to vanish;
defining $\alpha_{t}\equiv y_{t}^{2}/4\pi$ and $\alpha_{N,N^{\prime}}\equiv
g_{N,N^{\prime}}^{2}/4\pi$, and using the results of Machacek and Vaughn [10]
we can easily find the one-loop beta function for the high-energy theory:

$\frac{d\alpha_{t}}{d\ln\mu}=\frac{\alpha_{t}}{2\pi}\Big{[}\left(2N^{\prime}+N+M\right)\alpha_{t}-4C_{F}\alpha_{N}-4C_{F}^{\prime}\alpha_{N^{\prime}}\Big{]}~{}.$
(3.1)

The cancellation condition (2.4) with the more general gauge representations
of the top and Higgs fields becomes:

$0=2M\Lambda_{t}^{2}\alpha_{t}-3C_{F}\Lambda_{N}^{2}\alpha_{N}-3C_{F}^{\prime}\Lambda_{N^{\prime}}^{2}\alpha_{N^{\prime}}~{}.$
(3.2)

Here we also account for the fact that the cutoffs for the integrals in the
quadratic divergence are in general different, and denote them as
$\Lambda_{t}$, $\Lambda_{N}$, and $\Lambda_{N^{\prime}}$. In the context of
SUSY, $\Lambda_{t}$ is proportional to the stop mass, while the gauge cutoffs
are set by gaungino masses. To find a relation between the cutoffs, we would
need the details of the SUSY-breaking mechanism.

A simple model for producing squark and gaugino masses is gauge mediation
[14]. These models assume that there is Goldstino multiplet $X$ with a Yukawa
coupling to messenger charged under the SM gauge groups and aVEV

$\langle X\rangle=M_{UV}+\theta^{2}\mathcal{F}~{}.$ (3.3)

This yields gaugino masses at one-loop given by

$M_{N}=\frac{\alpha_{N}}{4\pi}N_{m}\frac{\mathcal{F}}{M_{UV}},\quad
M_{N^{\prime}}=\frac{\alpha_{N^{\prime}}}{4\pi}N_{m}^{\prime}\frac{\mathcal{F}}{M_{UV}}~{},$
(3.4)

with $N_{m}=2\sum T(R)$ the sum of indexes of the messengers. The stop masses
are

$\widetilde{m}_{R}^{2}=2C_{2}(t_{R})\frac{\alpha_{N}^{2}}{16\pi^{2}}N_{m}\left(\frac{\mathcal{F}}{M_{UV}}\right)^{2},\quad\widetilde{m}_{L}^{2}=2C_{2}(t_{L})\frac{\alpha_{N^{\prime}}^{2}}{16\pi^{2}}N_{m}^{\prime}\left(\frac{\mathcal{F}}{M_{UV}}\right)^{2}~{},$
(3.5)

so we expect the cutoffs to be

$\Lambda_{t}^{2}=\frac{1}{2}\left(\widetilde{m}_{R}^{2}+\widetilde{m}_{L}^{2}\right)\ln\left(\frac{\Lambda^{2}+\widetilde{m}_{t}^{2}}{\widetilde{m}_{t}^{2}}\right),\quad\Lambda_{N}^{2}=M_{N}^{2}\ln\left(\frac{\Lambda^{2}+M_{N}^{2}}{M_{N}^{2}}\right)~{},$
(3.6)

for a UV scale $\Lambda$, and an equivalent expression for
$\Lambda_{N^{\prime}}$. The logarithmic factors are equal to leading order in
the gauge coupling, and may be cancelled in (3.2), since the difference
corresponds to a higher loop effect. We will also assume that the gauge
coupling is at its fixed point up to the scale $\Lambda$; this may not be the
case in a fully realistic model, but again deviations from this limit
correspond to higher order loop effects [4].

In order to prevent a Higgs soft-mass at the 10 TeV scale, like the stop
squarks, a combination of chiral and vector-like messengers [18] may be needed
in the high energy theory; due to the opposite signs of their contributions to
the gaugino masses, a judicious choice of representations could allow for a
light Higgs mass. We are of course primarily concerned with contributions to
the Higgs mass from physics below the 10 TeV scale.

The new gauge coupling $\beta$ functions at two loops are:

$\displaystyle\frac{d\alpha_{N}}{d\ln\mu}$
$\displaystyle=-\frac{\alpha_{N}^{2}}{2\pi}\left(b_{N}+c_{N}\frac{\alpha_{N}}{4\pi}+d_{N}\frac{\alpha_{t}}{4\pi}+e_{NN^{\prime}}\frac{\alpha_{N^{\prime}}}{4\pi}\right)$
(3.7) $\displaystyle\frac{d\alpha_{N^{\prime}}}{d\ln\mu}$
$\displaystyle=-\frac{\alpha_{N^{\prime}}^{2}}{2\pi}\left(b_{N^{\prime}}+c_{N^{\prime}}\frac{\alpha_{N^{\prime}}}{4\pi}+d_{N^{\prime}}\frac{\alpha_{t}}{4\pi}+e_{N^{\prime}N}\frac{\alpha_{N}}{4\pi}\right)$
(3.8)

The coefficients in (3.7) and (3.8) are sensitive to the matter content of the
UV theory, and ensuring consistency between the three fixed point conditions
and the cancellation of the quadratic divergence requires specific choices of
representations and multiplicities of UV field content.

Figure 3: This figure shows the cutoff ratios versus $SU(N)$ gauge coupling
for various UV theories. The points in blue correspond to theories shown in
Table 2 that satisfy Eq. (3.9). The points in red correspond to theories shown
in Table LABEL:UVmatter3 that match the SM couplings, Eq. (3.10). The black
star represents an example theory whose cutoffs match a gauge-mediated SUSY-
breaking scenario.

At the scale $\Lambda_{IR}$ where $SU(N)\times SU(N^{\prime})\to SU(3)_{c}$,
the gauge couplings must obey

$\frac{1}{\alpha_{3}}\geq\frac{1}{\alpha_{N}}+\frac{1}{\alpha_{N^{\prime}}}~{}.$
(3.9)

There is tension between needing $\alpha_{N,N^{\prime}}>\alpha_{3}$ to satisfy
(3.9) and having the $\alpha_{t}\sim\alpha_{N,N^{\prime}}$ found via (3.1) or
(3.2) be small enough to run up to the SM value of the top Yukawa coupling.
Current measurements of the strong coupling constant, the top quark mass, and
Higgs VEV [15] yield

$\alpha_{3}^{\overline{MS}}(m_{Z})=0.1185\pm
0.51\%,\quad\alpha_{t}^{\overline{MS}}(m_{t})=0.06721\pm 5.65\%~{},$ (3.10)

so we may choose parameters such that (3.9) or (3.10) is satisfied, but not
both. Table 2 of the appendix summarizes theories with gauge couplings larger
than $\alpha_{3}$ at $\Lambda_{IR}=1$ TeV, while Table 3 summarizes those that
match the SM value of $\alpha_{t}$ at 1 TeV to within 5%. Figure 3 shows the
ratio of the top-loop cutoff to the $SU(N)$ gauge-loop cutoff for these
theories. We made the simplifying assumtion that
$\Lambda_{N}/\alpha_{N}=\Lambda_{N^{\prime}}/\alpha_{N^{\prime}}$ (or a
rational multiple thereof) as in gauge mediation. In the class of models
studied we generically require a rather large global $SU(M)$ symmetry for the
top quarks, or a large running of $\alpha_{N}$ or $\alpha_{N^{\prime}}$ from
the TeV scale to the 10 TeV scale.

While $t\bar{t}$ resonance experiments will be sensitive to additional gauge
symmetries, even greater experimental constraints come from precision
electroweak measurements [16], which would rule out the addition of many
additional $SU(2)_{L}$ doublets like our extra top quark multiplets unless
they have vector-like masses that do not require a Higgs VEV. Any solution to
the Higgs mass problem should also include a mechanism for sending
$\alpha_{2}$ to a fixed point as well, a feature we have ignored. In this
respect, a realistic model will probably look quite different from our toy
model.

##  4 Conclusions

We have seen that new gauge interactions that couple only to top quarks and
the Higgs field allow for a cancellation of the top-loop quadratic divergence
in the Higgs mass at one-loop. To enforce this condition, the SM must be
embedded in a UV theory (above a 5-10 TeV threshold) with fixed points for the
gauge couplings and the top Yukawa coupling. In a supersymmetric context, with
calculable stop and gaugino masses, this can lead to the cancellation of the
stop and gaugino mass dependence in the Higgs mass.

Since the cancellation depends on the top and gauge loop cutoffs, in order to
search for a fully realistic model one also has know the form of the cutoff.
For supersymmetric theories this means that one needs to include a mechanism
to mediate supersymmetry breaking to the supersymmetric SM sector. In this
sense the mechanism is highly constrained, which also means that it is highly
predictive.

We have not addressed how to account for the divergences from the
$SU(2)_{L}\times U(1)_{Y}$ gauge and Higgs quartic couplings. It is possible
that this could be addressed by additional fixed point conditions, or another
mechanism altogether; for example the theory could possess light (TeV scale)
winos, binos, and higgsinos. In order to preserve the fixed point structure
however, the $U(1)_{Y}$ would have to be embedded in a non-Abelian group.

In general, the cancellation of one-loop quadratic divergences in the Higgs
mass does not require the existence of IR fixed points, we can also envisage
that the ratios of couplings $\alpha_{N}/\alpha_{t}$ approaches a fixed
value222We thank Hsin-Chia Cheng for pointing out this possibility.. This is
much less restrictive than enforcing IR fixed points for both couplings
simultaneously, and provides hope that more realistic models can be
constructed.

## Acknowledgments

We thank Roni Harnik for suggesting the title. We thank Hsin-Chia Cheng, Csaba
Csáki, Markus Luty, Stephen Martin, and Martin Schmaltz for helpful
discussions. JT thanks the Aspen Center for Physics, where this work was
initiated. This research was supported in part by the DOE under grant DE-
SC-000999 and the NSF under grant PHY1066293.

##  Appendix A UV Matter Content

UV matter content and gauge parameters for theories with
$\alpha_{N,N^{\prime}}>\alpha_{3}$ at $\Lambda_{IR}=1$ TeV are given in Table
2. Here a gauge group of $SU(N)\times SU(N^{\prime})$ is assumed, with a
global symmetry group for top quarks of $SU(M)$. Weyl fermion numbers $n_{ij}$
of UV matter content are scanned over with $i,j=1,F,A$ corresponding to
singlets, fundamentals, or adjoints under $SU(N)$ and $SU(N^{\prime})$. We
assume SUSY in the UV theory to set the number of real scalars as
$s_{ij}=4n_{ij}$. Theories are listed in order of ascending relative error in
Eq. (3.10).

Tables 3 and LABEL:UVmatter3 list theories that match the running of
$\alpha_{t}$ to (3.10). Table 3 lists the top five minimal matter content
scenarios that satisfy Eq. (3.10). Table LABEL:UVmatter3 lists the theories
that match most closely with the gauge-mediated SUSY breaking scenario
described in Section 3.

All candidate theories have
$n_{A1},\,n_{1A},\,n_{FF},\,n_{AF},\,n_{FA},\,n_{AA}=0$ due to their large
impact on the $\beta$-functions of the couplings.

$N$ | $N^{\prime}$ | $M$ | $\alpha_{N}(\Lambda_{UV})$ | $\alpha_{N^{\prime}}(\Lambda_{UV})$ | $\alpha_{t}(\Lambda_{UV})$ | $n_{b_{N}}$ | $n_{b_{M}}$ | $n_{F1}$ | $n_{1F}$
---|---|---|---|---|---|---|---|---|---
$6$ | $6$ | $8$ | $.06602$ | $.06602$ | $.05925$ | $4$ | $4$ | $4$ | $0$
$4$ | $4$ | $4$ | $.06658$ | $.06658$ | $.06242$ | $4$ | $0$ | $3$ | $3$
$4$ | $4$ | $5$ | $.07053$ | $.07053$ | $.06225$ | $3$ | $2$ | $3$ | $1$
$6$ | $7$ | $9$ | $.06704$ | $.07631$ | $.06306$ | $5$ | $6$ | $2$ | $0$
$6$ | $8$ | $10$ | $.06821$ | $.08040$ | $.06444$ | $6$ | $8$ | $0$ | $0$
Table 2: UV Matter satisfying Eq. (3.9) that have the smallest error in Eq. (3.10) $N$ | $N^{\prime}$ | $M$ | $\alpha_{N}(\Lambda_{UV})$ | $\alpha_{N^{\prime}}(\Lambda_{UV})$ | $\alpha_{t}(\Lambda_{UV})$ | $n_{b_{N}}$ | $n_{b_{M}}$ | $n_{F1}$ | $n_{1F}$
---|---|---|---|---|---|---|---|---|---
$5$ | $4$ | $5$ | $.04055$ | $.07286$ | $.05199$ | $3$ | $0$ | $6$ | $1$
$4$ | $5$ | $7$ | $.07947$ | $.04340$ | $.04822$ | $3$ | $6$ | $1$ | $0$
$4$ | $5$ | $6$ | $.07538$ | $.04164$ | $.04825$ | $4$ | $4$ | $1$ | $2$
$4$ | $5$ | $5$ | $.07128$ | $.03987$ | $.04828$ | $5$ | $2$ | $1$ | $4$
$5$ | $4$ | $4$ | $.03865$ | $.06844$ | $.05202$ | $4$ | $2$ | $6$ | $1$
Table 3: Minimal UV Matter satisfying Eq. (3.10) Table 4:  UV matter content satisfying Eq. (3.10), listed in order of how closely these theories match a gauge-mediated SUSY breaking scenario N | N’ | M | $\alpha_{N}(\Lambda_{UV})$ | $\alpha_{N^{\prime}}(\Lambda_{UV})$ | $\alpha_{t}(\Lambda_{UV})$ | $n_{b_{N}}$ | $n_{b_{M}}$ | $n_{F1}$ | $n_{1F}$
---|---|---|---|---|---|---|---|---|---
8 | 8 | 4 | .04547 | .04547 | .05115 | 12 | 8 | 5 | 5
9 | 9 | 4 | .01336 | .07497 | .05065 | 14 | 10 | 7 | 4
10 | 10 | 9 | .01334 | .08300 | .04893 | 11 | 2 | 8 | 4
9 | 9 | 4 | .07497 | .01336 | .05065 | 14 | 10 | 4 | 7
8 | 9 | 10 | .08556 | .02270 | .04864 | 8 | 4 | 2 | 5
10 | 8 | 9 | .04486 | .05387 | .04962 | 7 | 2 | 10 | 1
10 | 7 | 9 | .08133 | .01133 | .05351 | 5 | 2 | 10 | 0
10 | 8 | 10 | .08361 | .008507 | .04971 | 6 | 0 | 8 | 3
6 | 5 | 7 | .06512 | .04314 | .05103 | 2 | 2 | 6 | 0
10 | 10 | 6 | .07631 | .02183 | .05398 | 14 | 8 | 4 | 7
9 | 8 | 7 | .01970 | .07990 | .05027 | 9 | 4 | 9 | 2
9 | 7 | 6 | .05623 | .03623 | .05160 | 8 | 6 | 9 | 1
8 | 10 | 9 | .08219 | .02859 | .05029 | 11 | 2 | 0 | 9
10 | 10 | 10 | .01453 | .08454 | .04904 | 10 | 0 | 8 | 4
10 | 9 | 10 | .06290 | .03867 | .05086 | 8 | 0 | 7 | 4
10 | 9 | 9 | .08212 | .01825 | .05271 | 9 | 2 | 6 | 5
8 | 9 | 6 | .02161 | .07734 | .05360 | 12 | 4 | 4 | 5
8 | 10 | 3 | .06343 | .03595 | .05519 | 17 | 10 | 0 | 10
8 | 6 | 2 | .07184 | .005005 | .05409 | 10 | 12 | 8 | 1
7 | 8 | 6 | .07228 | .02734 | .04903 | 10 | 2 | 2 | 8
8 | 7 | 10 | .08435 | .01834 | .04938 | 4 | 4 | 6 | 0
8 | 8 | 6 | .07658 | .02487 | .05326 | 10 | 4 | 4 | 6
8 | 9 | 7 | .02350 | .07889 | .05372 | 11 | 2 | 4 | 6
10 | 9 | 8 | .06033 | .03535 | .05064 | 10 | 4 | 7 | 4
9 | 8 | 4 | .05156 | .04232 | .05459 | 12 | 10 | 7 | 3
6 | 5 | 5 | .06157 | .03688 | .05107 | 5 | 2 | 6 | 2
7 | 6 | 4 | .07030 | .01650 | .05029 | 8 | 6 | 6 | 3
6 | 7 | 7 | .06399 | .04243 | .04920 | 7 | 2 | 2 | 5
10 | 10 | 10 | .08454 | .01453 | .04904 | 10 | 0 | 4 | 8
4 | 5 | 6 | .07538 | .04164 | .04825 | 4 | 4 | 1 | 2
10 | 9 | 4 | .03772 | .04939 | .05078 | 14 | 12 | 8 | 3
8 | 10 | 3 | .01164 | .07293 | .05249 | 17 | 10 | 2 | 8
8 | 6 | 4 | .07456 | .01109 | .05432 | 8 | 8 | 8 | 1
8 | 8 | 10 | .03137 | .08442 | .05364 | 6 | 4 | 6 | 0
9 | 7 | 7 | .08026 | .009409 | .05186 | 7 | 4 | 8 | 2
8 | 10 | 6 | .04275 | .05681 | .05289 | 14 | 4 | 1 | 9
6 | 8 | 6 | .01555 | .07655 | .04954 | 10 | 0 | 1 | 8
6 | 5 | 3 | .05802 | .03062 | .05110 | 7 | 6 | 6 | 2
8 | 9 | 8 | .02540 | .08045 | .05383 | 10 | 0 | 4 | 6
6 | 8 | 7 | .01835 | .07780 | .04963 | 9 | 2 | 1 | 6
9 | 8 | 6 | .01857 | .07775 | .05015 | 10 | 6 | 9 | 2
9 | 10 | 6 | .07652 | .02391 | .05239 | 14 | 6 | 2 | 9
7 | 9 | 5 | .06643 | .03463 | .05089 | 13 | 4 | 0 | 10

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