# 3D Scene Diffusion Guidance using Scene Graphs

Mohammad Naanaa Katharina Schmid 11footnotemark: 1 Yinyu Nie
Technical University of Munich
<EMAIL_ADDRESS>Equal contribution

###### Abstract

Guided synthesis of high-quality 3D scenes is a challenging task. Diffusion
models have shown promise in generating diverse data, including 3D scenes.
However, current methods rely directly on text embeddings for controlling the
generation, limiting the incorporation of complex spatial relationships
between objects. We propose a novel approach for 3D scene diffusion guidance
using scene graphs. To leverage the relative spatial information the scene
graphs provide, we make use of relational graph convolutional blocks within
our denoising network. We show that our approach significantly improves the
alignment between scene description and generated scene.

##  1 Introduction

Guided 3D scene generation is an emerging research area that has the potential
to revolutionize several domains, including computer graphics, virtual
reality, and gaming. By allowing users to generate 3D scenes with ease
precisely matching an input description, this technology can help streamline
the design process, enable more immersive experiences, and facilitate the
creation of realistic and complex simulations. Current state-of-the-art
methods [12, 21] already achieve promising results for 3D indoor scene
generation but often fall short in accurately aligning the generated scenes
with the input description. One possible reason for this limitation lies in
the reliance of these models solely on text embeddings of scene descriptions
for guidance. To address this issue, we hypothesize that conditioning on scene
graphs, which can capture more complex relationships between objects, will
yield superior results. With this motivation, we propose a novel diffusion
model that generates a 3D scene matrix, encoding a labeled bounding box
representation of the scene, guided by scene descriptions encoded as graphs.

The key contributions of the paper include:

  * •

We present a novel approach for 3D scene diffusion guidance using scene
graphs.

  * •

Our method includes a novel technique to condition matrix-shaped data on scene
graphs leveraging relational graph convolutional networks.

  * •

We demonstrate that our approach significantly improves the alignment of the
generated scenes with the given conditions.

##  2 Related Work

#### Diffusion Models

Diffusion models, also known as denoising diffusion probabilistic models
(DDPMs), have gained significant attention in the field of machine learning.

Diffusion models have been successfully applied to several generative tasks,
including natural language processing [9], text-to-speech [8], and text-to-
image like DALL-E [15] and Imagen [16], outperforming Generative Adversarial
Networks (GANs) on several image synthesis tasks [3]. These models provide a
powerful framework for probabilistic modeling and have inspired several recent
advancements in generative modeling. In the realm of 3D scenes, diffusion
models have received relatively less attention compared to their counterparts
in 2D applications. Existing works in the field have primarily focused on
generating single objects [10, 28, 30, 7, 29, 13].

#### 3D Scene Synthesis

Synthesizing 3D scenes poses unique challenges due to their higher level of
semantics, geometric complexity, and larger spatial extent. Existing
approaches for generating 3D scenes typically rely on various types of
generative models, including Variational Autoencoders (VAEs) [14], GANs [27],
and autoregressive models [11, 24, 12]. GANs are known for their ability to
produce high-quality samples, but they often suffer from limited diversity and
poor coverage of different modes in the data distribution. On the other hand,
VAEs have better mode coverage but struggle to generate faithful samples that
accurately represent the target distribution [25]. ATISS [12] demonstrated the
generative ability of autorgressive transformer architectures, synthesizing
new objects conditioned on previously generated ones step by step. While this
method allows to capture some inter-object relationships, it still falls short
in incorporating complex spatial dependencies between multiple objects.

For generating 3D indoor scenes with high fidelity and complexity, DiffuScene
[21] formulated the scene synthesis as a diffusion process. The method learns
a scene configuration by denoising a set of noisy matrices, where each row
encodes object properties. This explicit encoding of diverse scene
configurations into the denoising process enables generating faithful object
configurations that accurately represent the underlying distribution of scene
configurations. The authors also proposed a method to condition the generation
process on a textual scene description using extracted word embeddings from a
pre-trained BERT [2] encoder. While DiffuScene achieves state-of-the-art
results in unconditional scene synthesis, their guidance method often falls
short in accurately aligning the generated scenes with a complex input
description, as their approach allows for simple generation guidance from the
user’s perspective but lacks expressiveness for complex relationships between
objects.

#### Graph Conditioning

A graph is a data structure that captures the relationships (edges) between
entities (nodes). Each node in the graph contains associated features, and
graphs can be further specialized by introducing directionality to edges or
assigning relationship types to edges. The explicit representation of object
properties, spatial configurations, and semantic relationships has made graph
conditioning effective for generating diverse and contextually consistent data
for specific applications. HouseDiffusion [19] achieved competitive results in
generating floor plans using graph conditioning with 2D coordinates diffusion.
In their approach, an undirected graph is utilized, where nodes represent
rooms and edges represent door connections. Similarly, GeoDiff [26]
successfully demonstrated conditional 3D molecular generation by leveraging
molecular conformation graphs as guidance. In their setup, a molecular
conformation graph is used, comprising nodes representing atoms and edges
representing inner atomic bonds. While these methods have shown promising
results with relatively simple graph structures, to the best of our knowledge,
no existing work has explored conditioning the diffusion process on graphs
with directional edges and specified relation types. This presents an
interesting research gap that we aim to address in this paper.

##  3 Method

We propose a novel scene-graph-conditioned DDPM for generating 3D scenes.

###  3.1 3D Scene Diffusion

#### Scene Matrix

We represent every scene with a scene matrix - a 2D tensor of fixed size
$\mathbf{X}\in\mathbb{R}^{N\times D}$ comprising $N$ objects. Each object
$\mathbf{o}_{i}\in\mathbb{R}^{D}$ is defined by several attributes including
its centroid location $p\in\mathbb{R}^{3}$, three normalized axes
$n\in\mathbb{R}^{9}$ and its axis-aligned bounding box size
$s\in\mathbb{R}^{3}$. Since the number of objects varies across different
scenes, we incorporate an additional ’empty’ object and pad it into scenes,
thus achieving a fixed number of objects for each scene. This scene
representation allows us to directly retrieve a scene representation with
labelled bounding boxes.

#### Diffusion Process

The forward diffusion process is a pre-defined discrete-time Markov chain in
the dataspace $\mathcal{X}$, spanning all possible scenes
$\mathbf{X}\in\mathcal{X}$. Given a clean scene matrix $\mathbf{X}_{0}$ from
the underlying distribution $q(\mathbf{X}_{0})$, we gradually add Gaussian
noise to $\mathbf{X}_{0}$, obtaining a series of intermediate scene matrix
representations $\mathbf{X}_{1},...,\mathbf{X}_{T}$ with the same
dimensionality as $\mathbf{X}_{0}$, according to a pre-defined increasing
noise variance schedule $\beta_{1},...,\beta_{T}$ (with
$\beta_{1}<...<\beta_{T}$). The joint distribution
$q(\mathbf{X}_{1:T}|\mathbf{X}_{0})$ of the diffusion process can then be
expressed as:

$q(\mathbf{X}_{1:T}|\mathbf{X}_{0}):=\prod_{t=1}^{T}{q(\mathbf{X}_{t}|\mathbf{X}_{t-1})}$

where the diffusion step at time t is defined as:

$q(\mathbf{X}_{t}|\mathbf{X}_{t-1}):=\mathcal{N}(\mathbf{X}_{t};\sqrt{1-\beta_{t}}\mathbf{X}_{t-1},\beta_{t}\mathbf{I})$

A useful property of diffusion processes is that we can directly sample
$\mathbf{X}_{t}$ from $\mathbf{X}_{0}$ via the conditional distribution:

$q(\mathbf{X}_{t}|\mathbf{X}_{0}):=\mathcal{N}(\mathbf{X}_{t};\sqrt{\bar{\alpha_{t}}}\mathbf{X}_{0},(1-\bar{\alpha_{t}})\mathbf{I})$

where
$\mathbf{X}_{t}=\sqrt{\bar{\alpha_{t}}}\mathbf{X}_{0}+\sqrt{(1-\bar{\alpha_{t}})}\epsilon$
with $\alpha_{t}:=1-\beta_{t}$,
$\bar{\alpha_{t}}:=\prod_{r=1}^{t}{\alpha_{r}}$ and $\epsilon$ is the noise
used to corrupt $\mathbf{X}_{t}$.

#### Generative Process

Following the forward diffusion process, the generative denoising process is
parameterized as a Markov chain of learnable reverse Gaussian transitions.
Given a noisy scene matrix from a standard multivariate Gaussian distribution
$\mathbf{X}_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ as the initial state at
time step $T$, the generative process goes backwards and corrects
$\mathbf{X}_{t}$ obtaining a slightly less noisy version $\mathbf{X}_{t-1}$ at
each time step by using a learned Gaussian transition
$p_{\theta}(\mathbf{X}_{t-1}|\mathbf{X}_{t})$ parametrized by a learnable
network $\theta$. By repeating this reverse process until the maximum number
of steps $T$, we can reach the final state $\mathbf{X}_{0}$, resembling a
denoised scene matrix we aim to obtain. Specifically, the joint distribution
of the generative process $p_{\theta}(\mathbf{X}_{0:T})$ is formulated as:

$p_{\theta}(\mathbf{X}_{0:T}):=p(\mathbf{X}_{T})\prod_{t=1}^{T}{p_{\theta}(\mathbf{X}_{t-1}|\mathbf{X}_{t})}$

with

$p_{\theta}(\mathbf{X}_{t-1}|\mathbf{X}_{t}):=\mathcal{N}(\mathbf{X}_{t-1};\mu_{\theta}(\mathbf{X}_{t},t),\Sigma_{\theta}(\mathbf{X}_{t},t))$

where $\mu_{\theta}(\mathbf{X}_{t},t)$ and $\Sigma_{\theta}(\mathbf{X}_{t},t)$
are the predicted mean and covariance, respectively, of the Gaussian
$\mathbf{X}_{t-1}$ predicted from $\mathbf{X}_{t}$ by the denoising network
$\theta$. For simplicity, we set the variance to a constant
$\Sigma_{\theta}(\mathbf{X}_{t}):=\frac{1-\bar{\alpha_{t-1}}}{1-\bar{\alpha_{t}}}\beta_{t}$,
although Song et al. have shown that learnable covariances can improve the
generation quality [20]. We follow Ho et al. [5] and, instead of directly
predicting $\mu_{\theta}(\mathbf{X}_{t},t)$, we estimate the noise
$\epsilon_{\theta}(\mathbf{X}_{t},t)$ applied to perturb $\mathbf{X}_{t}$,
synthesizing more high-frequent details. In this approach,
$\mu_{\theta}(\mathbf{X}_{t},t)$ can then be reparametrized by substracting
the predicted noise according to Bayes’ theorem:

$\mu_{\theta}(\mathbf{X}_{t},t):=\frac{1}{\sqrt{\alpha_{t}}}(\mathbf{X}_{t}-\frac{\beta_{t}}{\sqrt{1-\bar{\alpha_{t}}}}\epsilon_{\theta}(\mathbf{X}_{t},t))$

#### Training Objective

In our DDPM scheduler implementation, the model can be trained to either
predict noise $\epsilon$ or $\mathbf{X}_{0}$ directly, using the
reparametrization trick. We refer to Ho et al. [5] for the details of the
derivation process.

When predicting $\mathbf{X}_{0}$, we utilize a custom loss function we
designed for 3D scene generation. This loss function combines multiple
components: (1) L2 loss, (2) a volume loss for 3D bounding boxes, and (3) an
aspect ratio loss. The incorporation of these loss components enhances the
quality and coherence of the generated scenes: the L2 loss ensures a
similarity of bounding box coordinates between generated and ground truth
scenes, the volume loss targets natural object volumes given their labels, and
the aspect ratio loss preserves natural object proportions.

$\mathcal{L}_{\text{total}}=\lambda_{1}\cdot\mathcal{L}_{l2}+\lambda_{2}\cdot\mathcal{L}_{\text{vol}}+\lambda_{3}\cdot\mathcal{L}_{\text{ratio}}$

For the additive noise prediction setting, the introduced custom losses are
not meaningful when computed on noise - neither 3D volume of predicted noise
nor its aspect ratio have any distinct meaning - and thus the model is
trainable using either L1 or L2 loss.

###  3.2 Scene Graph Guidance

We employ scene graphs as a conditioning input for our model to describe the
scene to be generated. Scene graphs offer the ability to capture complex
relationships between objects. We postulate that this approach is more
effective in expressing scene descriptions compared to the prevailing method
of solely conditioning on text embeddings.

The nodes of the scene graph condition correspond to the objects we aim to
generate within our scene. To extract the textual information associated with
these objects, we utilize a pre-trained FastText encoder [1] to generate text
embeddings for their respective labels, which offers the advantage of encoding
each label using a single token $c\in\mathbb{R}^{300}$. To establish
relationships between the objects, directed edges are employed to connect
specific nodes. Thereby, the relationship type associated with each edge
specifies the relative spatial positioning of the two respective objects.

Figure 1: Conditioning of the scene matrix on a scene graph. The scene matrix
is converted into a fully-connected graph representation with edges of neutral
relationship type (blue).

To condition the matrix-shaped scene matrix on a scene graph we follow a
series of steps outlined in Fig. 1. First, we convert the scene matrix into a
fully-connected graph representation, where each node corresponds to an object
embedding. During this conversion process, all edge connections are assigned a
neutral relationship type. To leverage the relational information between the
objects provided by the scene graph condition, we integrate the edge
connections along with their specified relationship type into the existing
edges of the graph scene representation.

Figure 2: Scene graph processing using Relational Graph Convolutional
Networks. The node ’table’ is updated based on all nodes connected to it via
incoming edges (’chair’, ’bed’, ’TV’). For each relationship type (’right’,
’fully connected’) a specific weight matrix is learned. Additionally, a self-
connection is employed.

To process the combined graph scene representation, we employ Relational Graph
Convolutional Network (RGCN) [18] blocks. RGCNs update each node based on the
nodes connected to it through incoming edges. This enables the network to
learn specific weight matrices for different relationship types describing the
relative positioning of the objects in the scene, as shown in Figure 2. By
incorporating spatial and semantic information from edge connections, the
network achieves more accurate updates of the objects’ bounding boxes. A self-
connection further enhances each node’s contextual awareness by considering
its current state. We found that fully connecting the graph scene
representation using a neutral relationship type yields significant benefits.
This approach enables the network to effectively update nodes, even when the
scene graph condition only provides sparse edge connections between nodes,
leading to more robust and comprehensive scene understanding.

###  3.3 Architecture

Figure 3: Architecture overview of our denoising network.

Fig. 3 provides an overview of the architecture we employ. In the initial
stage of our denoising network, we incorporate the time step embedding by
concatenating it with the input. Linear layers, complemented by a tanh
activation function, are then employed to reshape the data according to our
specific requirements. The core components of our denoising network are the
RGCN blocks. These blocks enable us to effectively process graph-shaped data,
leveraging the inherent relational information. Additionally, we incorporate
an attention block in our architecture. Here, we first inject the label
information from the condition by adding pooled embedding vectors to the
output of the last RGCN block. Subsequently, we simultaneously perform both
(1) cross-attention between the condition and the scene matrix to capture
label context and (2) self-attention to capture the global context of the
scene. To expedite the training process, we utilize skip connections as a
mechanism to facilitate information flow between different layers.

###  3.4 Classifier-Free Guidance

In order to enhance the alignment between the desired scene description and
our model’s output, we adopt the concept of classifier-free guidance, as
proposed by [6]. Thereby, the model is jointly trained in a conditional and
unconditional way by randomly ignoring the condition with a certain
probability. Adapting this concept to our problem requires extending the
approach to handle relational graph data. To ignore the label data, we follow
prior approaches and set the label information to zero [6]. Since our scene
graph condition also provides relational information, we must additionally
mask the respective edge connections, including their specific relationship
types. It is worth noting that this approach becomes feasible due to the
inherent fully connected neutral edge connections in the unconditioned graph
scene representation. This connectedness is essential for meaningful updates
in RGCNs, as they rely on edge connections to generate coherent and meaningful
outputs in the unconditional case.

The adoption of classifier-free guidance significantly impacted the
architecture of our model. To successfully scale the output towards the
influence of the condition, it is essential that the model can generate
meaningful outputs in an unconditional way. During unconditional training,
setting the label information to zero causes the cross-attention layer to
produce a zero output. To address this limitation and enable the learning of
meaningful outputs in the unconditional scenario, we introduced an alternative
information flow. Specifically, we incorporated the cross-attention block in
parallel with a self-attention block and added their results, enabling
information flow in the unconditional case.

During the inference step we observe that high guidance weights enhance the
description-scene alignment. However, this improvement comes at the cost of
lower fidelity of data. We attribute this decrease in fidelity to the higher
guidance weights causing the predicted values $\mathbf{X}_{0}$ to surpass the
bounds of the training data, which is typically in the range of [-1, 1]. To
address this issue, we employ dynamic thresholding as a solution, following
[16].

##  4 Experiments

Implementing our proposed method, we prepare the required dataset and perform
a series of architectural experiments.

###  4.1 Dataset

#### Data Sources

Our method requires a dataset containing both 3D scenes and their
corresponding scene graphs for training. We build our experiments on two
datasets: The underlying dataset is (1) 3RScan [22] \- a dataset featuring
1482 RGB-D scans of 478 scenes across multiple time steps. Building on top of
it, (2) 3DSSG [23] is a semi-automatically generated dataset that contains
semantically rich scene graphs of 3D scenes featuring 1482 samples with 48k
object nodes and 544k edges. The authors of the original paper proposed a
learned method that regresses a scene graph from the point cloud of a scene.
Combining both datasets, we get 3D scene scans from 3RScan including several
objects with ground truth annotations of object labels and bounding boxes, as
well as corresponding scene graphs from 3DSSG describing present objects and
their relationships.

#### Data Filtering

As a real-world point cloud scan dataset, 3RScan contains inherent noise and
variations, which are further amplified during the semi-automatic scene graph
generation conducted by 3DSSG. In order to ensure the quality and reliability
of our experimental results, we performed a careful filtering process on the
dataset, applying several filtering criteria to refine the dataset for our
purposes.

Firstly, we filtered out scenes with particularly high noise levels from the
original set of 478 environments. Real-world datasets often contain artifacts,
bounding box collisions, or incomplete data, which can adversely affect the
learning process. By excluding scenes with excessive noise, we aimed to ensure
a higher level of consistency and reliability in our dataset. This step was
achieved with visual inspection of available ground truth 3D samples.

Furthermore, we reduced the unnecessary complexity of scene graphs by only
keeping 23 available spatial relationships, removing visual and functional
relationships. This approach allowed us to focus on capturing the spatial
arrangement of objects, which is crucial for our research contribution goal.

Additionally, to tackle the issue of class imbalance, we opted to retain only
the most frequent object categories in the dataset. Real-world scenes often
exhibit a wide range of object categories, with some categories being rarely
represented. By keeping only the most frequent 51 object categories, we aimed
to mitigate the imbalance in the dataset, ensuring a more representative
distribution of object types for training our model.

Lastly, following the architectural design of our scene matrix from 3.1, we
set the maximum number of representable objects to $N=20$. We obtained this
value empirically as it allows for object-rich scenes but also normalizes
outlier scenes containing a lot of small objects that might disproportionately
affect the learning process or introduce unnecessary complexity. We filtered
out these excessive objects keeping their total per-scene count below 20.

It is important to note that after applying these filtering steps, there were
cases where scenes contained no objects or had no relations left. Such cases
were also excluded from our final dataset as they provided no meaningful
information for our diffusion process.

#### Final Curated Dataset

As a result of the filtering steps, we ended up with a curated dataset
consisting of 407 aggregated scenes. This refined dataset ensures a more
reliable and balanced foundation for evaluating the performance of our
conditioning approach for diffusion process in 3D scene generation.

###  4.2 Architectural Ablation Studies

We design a series of ablation studies to investigate the impact of various
design choices on the performance of our scene-graph-conditioned DDPM. We
focused on understanding how different architectural components contribute to
the generation quality and coherence of the generated 3D scenes.

#### Cross-Attention

The cross-attention block plays a significant role in incorporating label
information from the scene graph condition into our model. This enables the
model to consider the semantic meaning of the label for the size of the
respective bounding box during scene generation. We observed that the cross-
attention mechanism is especially useful in ensuring that the generated
objects have correct volumes. By incorporating information about object
labels, the model can better align the generated shapes with the desired
semantics, leading to more coherent and meaningful 3D scenes.

#### Self-Attention

Incorporating self-attention in our architecture enhances the overall fidelity
of the model. Self-attention helps to address issues such as bounding box
collisions, ensuring that the generated objects are spatially distributed in a
more realistic manner. Additionally, the self-attention mechanism enables the
model to capture global context, which is crucial for maintaining consistency
and coherence between different objects in the entire scene.

#### Time Step Injection

During our experiments, we explored two approaches for injecting the time step
information into the model. The first approach incorporates the time step
embedding through an element-wise addition with the input. The second approach
involves concatenating the time step embedding with the input at the beginning
of the network. We found that the second approach, using concatenation at the
beginning, was more effective. This suggests that considering the time step
information as an independent factor from the start allows the model to better
leverage this temporal information when predicting noise in the current data.

#### Skip Connection

We incorporated skip connections in our architecture to facilitate enhanced
information flow between different layers. The presence of skip connections
improved the gradient flow and accelerated the convergence during training,
enabling the model to learn accurate updates for scene objects more easily.
This architectural choice helped enhance the stability and overall performance
of the model during the scene generation process.

###  4.3 Implementation

#### Loss

Experimentally, we observed marginally better performance when training to
predict noise and followed this setting for training our best performing
model. For the diffusion process, we tried both linear and cosine schedule and
observed a better results with the linear schedule. To increase the stability
of denoising training, we normalize the scene matrices to values between
$[-1,1]$.

#### Classifier-Free Guidance

For the classifier-free guidance, we have found the condition drop probability
$=15\%$ and condition scale $=3$ [6] to achieve the best mode coverage vs.
sample fidelity trade-off performance. To balance the effect of high guidance
weights on the prediction value distribution, we apply dynamic thresholding
with an 85th percentile.

#### RGCN

For RGCN blocks, we observed $\texttt{mean}(\cdot)$ to be the best aggregation
function; regularization with basis-decomposition [18] using 4 bases further
helped dealing with overfitting on rare relations in RGCN layers.

#### Training

We train our scene diffusion model on a single Nvidia A10G 24GB GPU with a
batch size of 128 for 10000 epochs. We follow the proposed 80/10/10 dataset
split by the original dataset authors, slightly rebalancing the splits after
the scenes filtering mentioned in 4.1. We optimize using Adam with learning
rate initialized to $lr=5e-4$ and weight decay of $1e-4$ with an additional
$lr$ reduction on plateau by a $0.8$ factor following $60$ patience threshold.

##  5 Evaluation

We postulated that conditioning the diffusion process on scene graphs leads to
enhanced positional alignment since scene graphs provide relative spatial
information within the the edges and relationships. We hypothesized that
utilizing RGCNs would enable our model to incorporate this relational
information into the generated data.

To evaluate this hypothesis, we present both qualitative and quantitative
results, showcasing the effectiveness of our method.

###  5.1 Quantitative Results

#### Relationship Alignment Score (RAS)

In order to quantify and evaluate the alignment between the generated scenes
and the scene graphs used as conditions in our generative process, we propose
a new metric called the Relationship Alignment Score (RAS). The RAS measures
the extent to which the generated scenes capture the relationships specified
in the scene graphs.

The calculation of the RAS involves inspecting each relationship $r$ between
two objects and determining if the corresponding relation is present in the
generated scene or not, assigning it a score $\delta(r)$ of 1 or 0,
respectively. To compute the RAS for a complete generated scene $s$, we
calculate the average value of scores for all relationships $r\in R_{s}$ from
the scene graph used for guidance. This average value between 0 and 1
indicates the level of alignment between the generated scene and the specified
relationships in the scene graph, with higher value denoting a better overall
alignment. By computing the RAS for every generated scene $s\in S$ and
averaging the scores, we obtain an overall measure of how well the generative
process aligns with the specified relationships in the scene graphs.

$\text{{RAS}}=\frac{1}{|S|}\cdot\sum_{s\in S}(\frac{1}{|R_{s}|}\cdot\sum_{r\in
R_{s}}\delta(r))$

$\delta(r)=\begin{cases}1,&\text{if relationship }r\text{ holds}\\\
0,&\text{otherwise}\end{cases}$

The use of the RAS metric allows us to quantitatively assess the ability of
our approach to generate scenes which adhere to the specified spatial
relationships and semantics defined by the scene graphs. In our experiments,
we report the average RAS across all generated scenes to provide a
comprehensive evaluation of the alignment with the scene graphs.

#### Relationship Information Injection

Utilizing the alignment score to assess whether our model is able to
effectively make use of spacial relational information provided by the scene
graphs, we devised a comparative study between two models. The baseline model
considered solely the label information of the condition without incorporating
any relational information, whereas the second model was conditioned on the
entire scene graph, including edge and relationship information.

Our findings revealed that the baseline model performed worse and yielded a
lower RAS value of 44%, compared to the model trained with relational
information injection with an alignment score of 67%, both evaluated on 45
test conditions. This outcome strongly supports our hypothesis that scene
graphs offer valuable spatial information which the denoising network can
effectively exploit using RGCNs, improving the scene alignment with the
condition. This result is also confirmed visually, when comparing the
generated scenes of both models for the same condition in Figure 4.

Figure 4: Two synthesized 3D scenes (viewed from top) with the same scene
graph condition seen on the right. The top scene was generated by a model
incorporating relational information in its RGCN blocks whereas the bottom
scene was produced by a model trained without relational information
injection. A much better alignment between the scene graph condition and the
actual scene is observed by the prior model, matching its higher RAS value.

###  5.2 Qualitative Results

The qualitative evaluation of our proposed method is depicted in Figure 5. We
observe that the generated scenes align well with the specified scene graphs.
For each row in the figure, we provide a natural language description of the
scene and the corresponding scene graph that served as the input for the
generative process. The subsequent columns display the synthesized 3D scenes,
both from the side and top views. We carefully selected results to showcase
the effectiveness of our approach across various scenarios, from simple scenes
to scenes with higher complexity in object count as well as convoluted
relational patterns. Furthermore, even for sparse graph data (e.g. graphs
consisting of multiple disconnected subgraphs), our method achieves meaningful
results consistent with the description.

The 3D scenes accurately capture the relative spatial positioning of objects
and maintain semantic consistency with the given descriptions. Furthermore,
the qualitative assessment confirms that the incorporation of relational
information using RGCNs significantly enhances the model’s ability to align
the scenes with the specified relationships and to achieve realistic and
contextually appropriate 3D scene synthesis.

Overall, the qualitative evaluation demonstrates that our proposed method is
effective in generating 3D scenes that align with the input scene graphs,
providing a powerful tool for guided 3D scene synthesis.

Figure 5: An overview of 3D scene synthesis results. Each row depicts a single
result: first column displays a natural language description of a scene,
second column shows a corresponding scene graph used as input for the
generative process. The remaining two columns depict the synthesized 3D scenes
both from the side and top views. The selected results display generative
results for (1) very complex, (2) disconnected, (3) repetitive, and (4) simple
scene graphs.

##  6 Limitations and Outlook

While our method has demonstrated impressive results in creating 3D scenes
that conform to the spatial relationship requirements of the condition, it
does have certain limitations. Firstly, the use of fixed-shaped scene matrices
restricts the number of objects within a scene to a pre-defined maximum (e.g.,
20). Consequently, our method is unable to synthesize large-scale scenes
beyond that limit. Secondly, in our efforts to validate the effectiveness of
our approach in synthesizing scenes which accurately incorporate relative
positioning of objects based on the condition, we focused solely on
synthesizing labeled bounding boxes. However, for more realistic-looking
scenes, we suggest to expand the generated scene matrices to include a shape
code $f\in\mathbb{R}^{F}$ for each object, as proposed by [21]. By leveraging
the shape code, along with the bounding box size and object label, we can
retrieve the most similar CAD model from a 3D furniture dataset, such as
3D-FUTURE [4], for each object in the generated scene matrix.

The main limitation of our work is caused by the scarcity of paired scene and
scene graph data. Our model was trained on a relatively small dataset of less
than 400 samples, which significantly limits its ability to generalize. This
limitation can only partly be addressed through various data augmentation
techniques. We postulate that the limited number of training samples is
responsible for our model’s better performance in predicting noise $\epsilon$
as opposed to predicting $\mathbf{X}_{0}$. The model’s inability to overfit
easily when predicting random noise contributes to this behavior. Regrettably,
predicting $\mathbf{X}_{0}$ would have enabled us to employ a more
sophisticated loss function, incorporating a volume loss for 3D bounding boxes
and an aspect ratio loss as described in 3.1. However, as predicting the noise
led to better results on the small dataset, we could only utilize a simple
reconstruction loss. To leverage the advantages of both prediction methods
while dealing with the constraints of a limited dataset, we propose an
approach to predict both $\mathbf{X}_{0}$ and $\epsilon$, and subsequently
merge the predictions by interpolating between predicting $\mathbf{X}_{0}$
directly and predicting via $\epsilon$ [17]. By doing so, we aim to maintain
high variance in the training process while applying a more comprehensive loss
function, thus enhancing the realism of the generated scenes. While previous
work by [17] demonstrated that predicting $\mathbf{X}_{0}$ and $\epsilon$ in
parallel is stable and even yields slightly better fidelity scores compared to
the standard method of predicting $\epsilon$ only, we leave further
experimentation with this method, and its potential for overcoming the
challenges posed by a restricted dataset, to future research endeavors.

Finally, in case of an increased availability of training data it is a
promising direction to experiment with more complex architectures. Utilizing a
larger model holds the potential to achieve significantly improved fidelity
scores. As such, an expanded dataset combined with a sophisticated
architecture represents an exciting avenue to further elevate the performance
of our approach.

##  7 Conclusion

In this paper, we have addressed the challenge of accurately aligning
generated scenes with input descriptions in the context of 3D scene generation
using denoising diffusion probabilistic models. We presented a novel approach
using scene graphs with directional edges and specified relationship type to
guide the diffusion process, elevating the level of complexity the scene
descriptions can encode. To successfully incorporate spatial information from
the scene graph condition into the generated scenes we employ relational graph
convolutional blocks. We introduced the RAS metric, providing a quantitative
tool for evaluating the spatial alignment of generated scenes and consequently
enabling the assessment of the effectiveness of our proposed method.
Throughout our experiments and evaluations, we have demonstrated the efficacy
of scene graphs in encoding relative spatial positioning information and the
effectiveness of our approach in leveraging this information to achieve
significant improvements in scene alignment with the given conditions. By
effectively addressing the spatial alignment challenge, our novel approach
makes a substantial contribution to the field of guided 3D synthesis. We
believe that the integration of scene graph information holds immense
potential for advancing the capabilities of guided 3D scene synthesis. As
such, we hope our work will inspire future research in the realm of scene
graph guidance.

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#  On scalable $K$-frames and a version of Lax-Milgram theorem

F. Javadi1 and M. J. Mehdipour1111Corresponding author 1 Department of
Mathematics, Shiraz University of Technology, P. O. Box $71555-313$, Shiraz,
Iran.

###### Abstract

In this paper, we first prove a theorem by a little modification on the Lax-
Milgram theorem. Then, using $K$-frames, we obtain lower and upper bounds for
the results obtained from this theorem. Also, we present some methods for the
characterization of scalable $K$-frames. Finally, we introduce piecewise
scalable $K$-frames and give necessary and sufficient conditions for a
$K$-frame to be piecewise scalable.

Keywords: Lax-Milgram theorem, parseval $K$-frame, scalable $K$-frame,
piecewise scalable $K$-frame.
2020 MSC: 42C15

##  1 Introduction

Let $H$ denote a separable infinite dimensional Hilbert space and ${\cal
L}(H)$ be the space of all bounded linear operators from $H$ into $H$. A
sequence $\\{f_{j}\\}_{j\in\mathbb{Z}}$ is called a frame, if there exist
constants $A,B>0$ such that for every $f\in H$,

$A\,\|f\|^{2}\leq\sum_{j\in\mathbb{Z}}|\langle f,f_{j}\rangle|^{2}\leq
B\,\|f\|^{2};$ (1.1)

see [7] for more details. If $A=B=1$, then we call
$\\{f_{j}\\}_{j\in\mathbb{Z}}$ a Parseval frame. A frame
$\\{f_{j}\\}_{j\in\mathbb{Z}}$ is called a scalable frame if there exist
constants $\\{c_{j}\\}\in\ell^{\infty}(J)$ such that
$\\{c_{j}f_{j}\\}_{j\in\mathbb{Z}}$ is a Parseval frame. So for every $f\in
H$, we have

$\displaystyle f=\sum_{j\in\mathbb{Z}}\langle
f,c_{j}f_{j}\rangle\,c_{j}f_{j}\quad\text{and}\quad\sum_{j\in\mathbb{Z}}|\langle
f,c_{j}f_{j}\rangle|^{2}=\|f\|^{2}.$

In this paper, we always assume that $K\in\mathcal{L}(H)$. A sequence
$\\{f_{j}\\}_{j\in\mathbb{Z}}$ is called a $K$-frame, if there exist constants
$A,B>0$ such that for every $f\in H$,

$A\,\|K^{*}f\|^{2}\leq\sum_{j\in\mathbb{Z}}|\langle f,f_{j}\rangle|^{2}\leq
B\,\|f\|^{2}.$ (1.2)

If

$\|K^{*}f\|^{2}=\sum_{j\in\mathbb{Z}}|\langle f,f_{j}\rangle|^{2},$

then $\\{f_{j}\\}_{j\in\mathbb{Z}}$ is called a Parseval $K$-frame on $H$. The
operator $T:H\rightarrow\ell^{2}(J)$ defined by

$Tf=\\{\langle f,f_{j}\rangle\\}_{j\in\mathbb{Z}},$

is called analysis operator of $\\{f_{j}\\}_{j\in\mathbb{Z}}$. The operator
$S:H\rightarrow H$ defined by

$Sf=T^{*}Tf=\sum_{j\in\mathbb{Z}}\langle f,f_{j}\rangle f_{j},$

is called frame operator of $\\{f_{j}\\}_{j\in\mathbb{Z}}$; see [9, 10, 11,
12, 15].

It’s clear that if $K$ is the identity map on $H$, then every $K$-frame is an
ordinary frame. Hence, $K$-frames arise naturally as a generalization of the
ordinary frames. Parseval $K$-frames are one of the most important types of
$K$-frames. They are applied in signal communication. It is natural to ask how
to make a Parseval $K$-frame from a given $K$-frame. Kutyniok et. al [13]
solved this question for frames and introduced the conception of scalable
frames. This notion has been investigated by some authors [2,4,5,6,14]. These
results cause us to investigate the conception of scalable $K$-frames.

Several authors studied scalable frames and piecewise scalable frames. For
example, Cahill and Chen [2] investigated the following question: Under what
conditions is a frame scalable? Casazza et. al [3] generalized the notion of
scalable frames and introduced piecewise scalable frames. They also
characterized them. In this paper, we continue these investigations.

The outline of the paper is as follows. In Section 2, we prove an analogue of
Lax-Milgram theorem and give an application of it for $K$-frames. In Section
3, we construct a scalable $K$-frame by a given scalable and a bounded
operator. In Section 4, we introduce piecewise scalable $K$-frame and
characterize them.

##  2 Generalization of Lax-Milgram Theorem

Let ${\cal H}$ be a real Hilbert space and $\sigma:{\cal H}\times{\cal
H}\rightarrow\mathbb{R}$ be a bilinear map. For $f_{0}\in{\cal H}$ and a non-
empty closed convex subset $C$ of ${\cal H}$, we define the function
$J_{f_{0},\sigma}:C\rightarrow{\mathbb{R}}$ by

$\displaystyle J_{f_{0},\sigma}(v)=\frac{1}{2}\sigma(v,v)-\langle
KK^{*}f_{0},v\rangle.$

Let us recall that $\sigma$ is called coercive if there exists a constant
$\alpha>0$ such that for every $v\in{\cal H}$

$\displaystyle\sigma(v,v)\geq\alpha\,\|v\|^{2},$

and is called continuous if there exists $\beta>0$ such that for every
$u,v\in{\cal H}$

$\displaystyle|\sigma(u,v)|\leq\beta\,\|u\|\,\|v\|.$

Also, a self-adjoint operator $A:{\cal H}\rightarrow{\cal H}$ is called
coercive if there exists a constant $\alpha>0$ such that for every $v\in{\cal
H}$

$\displaystyle\langle Av,v\rangle\geq\alpha\,\|v\|^{2}.$

The following result is a little modification of the Stampacchia and the Lax-
Milgram theorems [1].

######  Theorem 2.1.

Let $\sigma$ be a continuous coercive bilinear map on ${\cal H}$, and $C$ be a
non-empty closed convex subset of ${\cal H}$. Then for every bounded linear
functional $L$ on ${\cal H}$, there exist unique $u_{0}\in C$ and
$f_{0}\in{\cal H}$ such that for every $v\in{\cal H}$

$\displaystyle\sigma(u_{0},v-u_{0})\geq\langle
KK^{*}f_{0},v-u_{0}\rangle\quad\emph{and}\quad L(v)=\langle f_{0},v\rangle.$

Moreover, if $\sigma$ is symmetric, then $u_{0}$ is characterized by the
property

$\displaystyle J_{f_{0},\sigma}(u_{0})=\min_{v\in C}J_{f_{0},\sigma}(v).$

###### Proof.

Let $L$ be a bounded linear functional on ${\cal H}$. Using the Riesz
representation theorem, there exists a unique element $f_{0}\in{\cal H}$ such
that for every $v\in{\cal H}$,

$\displaystyle L(v)=\langle f_{0},v\rangle.$

Put $\tilde{L}:=LKK^{*}$. Then there exists $f_{0}^{\prime}\in{\cal H}$ such
that for every $v\in{\cal H}$, we have $\tilde{L}(v)=\langle
f_{0}^{\prime},v\rangle$. So for every $v\in{\cal H}$, we obtain

$\displaystyle\langle f_{0}^{\prime},v\rangle$ $\displaystyle=$
$\displaystyle\tilde{L}(v)=LKK^{*}(v)$ $\displaystyle=$ $\displaystyle
L(KK^{*}(v))$ $\displaystyle=$ $\displaystyle\langle f_{0},KK^{*}v\rangle$
$\displaystyle=$ $\displaystyle\langle KK^{*}f_{0},v\rangle.$

For every $u\in{\cal H}$, the linear functional $v\mapsto\sigma(u,v)$ is
continuous. So, we find some unique element in ${\cal H}$, denoted by $\Lambda
u$, such that $\sigma(u,v)=\langle\Lambda u,v\rangle$ for every $v\in{\cal
H}$. Then the operator $\Lambda$ is linear and there exists a constant
$\beta>0$ such that for every $u\in{\cal H}$

$\displaystyle\|\Lambda u\|^{2}$ $\displaystyle=$
$\displaystyle|\langle\Lambda u,u\rangle|$ $\displaystyle=$
$\displaystyle|\sigma(u,u)|\leq\beta\,\|u\|^{2}.$

Since $\sigma$ is coercive, there exists a constant $\alpha>0$ such that for
every $u\in{\cal H}$

$\displaystyle\langle\Lambda u,u\rangle=\sigma(u,u)\geq\alpha\|u\|^{2}.$ (2.2)

Set $\gamma:=\alpha/\beta^{2}$ and define the map $T$ on $C$ by

$\displaystyle T(v)=P_{C}(\gamma KK^{*}f_{0}-\gamma\Lambda v+v),$

where $P_{C}$ is the projection onto the closed convex set $C$. Note that
every projection on $C$ is not increase distance; see Proposition 5.3 in [1].
Thus

$\displaystyle\|Tv_{1}-Tv_{2}\|$ $\displaystyle=$ $\displaystyle\|P_{C}(\gamma
KK^{*}f_{0}-\gamma\Lambda v_{1}+v_{1})-P_{C}(\gamma KK^{*}f_{0}-\gamma\Lambda
v_{2}+v_{2})\|$ $\displaystyle\leq$ $\displaystyle\|(\gamma
KK^{*}f_{0}-\gamma\Lambda v_{1}+v_{1})-(\gamma KK^{*}f_{0}-\gamma\Lambda
v_{2}+v_{2})\|$ $\displaystyle\leq$
$\displaystyle\|(v_{1}-v_{2})-\gamma(\Lambda v_{1}-\Lambda v_{2})\|.$

So, using (2) and (2.2), we obtain

$\displaystyle\|Tv_{1}-Tv_{2}\|^{2}$ $\displaystyle=$
$\displaystyle\|(v_{1}-v_{2})\|^{2}-2\gamma\langle\Lambda v_{1}-\Lambda
v_{2},v_{1}-v_{2}\rangle+\gamma^{2}\|\Lambda v_{1}-\Lambda v_{2}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|(v_{1}-v_{2})\|^{2}(1-2\gamma\alpha+\gamma^{2}\beta^{2}).$

Hence $T$ is a strict contraction. In view of Theorem 5.7 in [1], $T$ has a
unique fixed point, i.e., there exists a unique $u_{0}\in C$ such that

$\displaystyle u_{0}=T(u_{0})=P_{C}(\gamma KK^{*}f_{0}-\gamma\Lambda
u_{0}+u_{0}).$

This implies that

$\displaystyle\langle\gamma KK^{*}f_{0}-\gamma\Lambda
u_{0}+u_{0}-u_{0},v-u_{0}\rangle\leq 0;$

or equivalently,

$\displaystyle\langle\Lambda u_{0},v-u_{0}\rangle\geq\langle
KK^{*}f_{0},v-u_{0}\rangle.$

Therefore, $\sigma(u_{0},v-u_{0})\geq\langle KK^{*}f_{0},v-u_{0}\rangle$.

Now, let $\sigma(u,v)$ be symmetric. Then the mapping

$(u,v)\mapsto\sigma(u,v)=\langle\Lambda u,v\rangle$

defines an inner product on ${\cal H}$. With this inner product, ${\cal H}$ is
also a Hilbert space. By the Riesz theorem, there exists a unique element
$g_{0}\in{\cal H}$ such that for every $v\in{\cal H}$

$\displaystyle\tilde{L}(v)=\langle KK^{*}f_{0},v\rangle=\sigma(g_{0},v).$

The rest of the proof is similar to the proof of Theorem Stampacchia; see
Theorem 5.6 in [1]. So we omit it. ∎

Let $\sigma(u,v)$ be a continuous bilinear on ${\cal H}$. Then for every
$v\in{\cal H}$, we have

$\displaystyle\sigma(v,v)\geq 0.$

Thus the function $v\mapsto\sigma(v,v)$ is convex. This fact together with
Theorem 2.1 proves the following result.

######  Corollary 2.2.

Let $\sigma(u,v)$ be a continuous coercive bilinear on ${\cal H}$. Then for
every bounded linear functional $L$ on ${\cal H}$, there exist unique
$u_{0},f_{0}\in{\cal H}$ such that for every $v\in{\cal H}$

$\displaystyle\sigma(u_{0},v-u_{0})=\langle
KK^{*}f_{0},v-u_{0}\rangle\quad\emph{and}\quad L(v)=\langle f_{0},v\rangle.$

Moreover, if $\sigma$ is symmetric, then $u_{0}$ is characterized by the
property

$\displaystyle J_{f_{0},\sigma}(u_{0})=\min_{v\in C}J_{f_{0},\sigma}(v).$

Let $\\{f_{j}\\}_{j\in\mathbb{Z}}$ be a $K$-frame for ${\cal H}$ with the
frame operator $S$. Define the bilinear form $\sigma_{S}:{\cal H}\times{\cal
H}\rightarrow\mathbb{C}$ by

$\displaystyle\sigma_{S}(u,v)=\langle Su,v\rangle.$ (2.3)

Then $\sigma_{S}$ is bounded, coercive and symmetric. So Corollary 2.2 yields
the following:

######  Corollary 2.3.

Let $\\{f_{j}\\}_{j\in\mathbb{Z}}$ be a $K$-frame for ${\cal H}$ with the
frame operator $S$. Then for every bounded linear functional $L$ on ${\cal
H}$, there exist unique $u_{0},f_{0}\in{\cal H}$ for which

$\displaystyle\langle Su_{0},v\rangle=\langle
KK^{*}f_{0},v\rangle\quad\emph{and}\quad L(v)=\langle f_{0},v\rangle$

for all $v\in H$. Furthermore,

$\displaystyle J_{f_{0},\sigma_{S}}(u_{0})=\min_{v\in
C}J_{f_{0},\sigma_{S}}(v).$

In the following, using $K$-frames, we get lower and upper bounds for
$\min_{v\in{\cal H}}J_{f_{0},\sigma_{S}}$.

######  Theorem 2.4.

Let $\\{f_{j}\\}_{j\in\mathbb{Z}}$ be a $K$-frame for ${\cal H}$ with the
frame operator $S$ and $K$-frame constants $A<B$. If $f_{0}\in{\cal H}$, then

$\displaystyle-\frac{\|K^{*}f_{0}\|^{2}}{2A}\leq\min_{v\in{\cal
H}}J_{f_{0},\sigma_{S}}(v)\leq-\frac{7}{32}\frac{\|K^{*}f_{0}\|^{2}}{B\|f_{0}\|^{2}}\leq-\beta\|K^{*}f_{0}\|^{2},$

for some $\beta>0$.

###### Proof.

Let $\\{f_{j}\\}_{j\in\mathbb{Z}}$ be a $K$-frame for $\cal{H}$. By (1.2), for
every $v\in\cal{H}$, we have

$\displaystyle J_{f_{0},\sigma_{S}}(v)=\bigg{(}\frac{1}{2}\langle
Sv,v\rangle-\langle KK^{*}f_{0},v\rangle\bigg{)}$ $\displaystyle\geq$
$\displaystyle\bigg{(}\frac{A}{2}\|K^{*}v\|^{2}-\langle
KK^{*}f_{0},v\rangle\bigg{)}.$

Apply Cauchy-Schwartz inequality to conclude that

$\displaystyle\langle KK^{*}f_{0},v\rangle=\langle
K^{*}f_{0},K^{*}v\rangle\leq\|K^{*}f_{0}\|\|K^{*}v\|.$

So we have

$\displaystyle
J_{f_{0},\sigma_{S}}(v)\geq\bigg{(}\frac{A}{2}\|K^{*}v\|^{2}-\|K^{*}f_{0}\|\|K^{*}v\|\bigg{)}.$

The minimum of the function
$v\mapsto\frac{A}{2}\|K^{*}v\|^{2}-\|K^{*}f_{0}\|\|K^{*}v\|$ is attained in
point $\frac{\|K^{*}f_{0}\|}{A}$, and hence

$\displaystyle\min_{v\in{\cal H}}J_{f_{0},\sigma_{S}}(v)$ $\displaystyle\geq$
$\displaystyle\bigg{(}\frac{A}{2}\frac{1}{A^{2}}\|K^{*}f_{0}\|^{2}-\|K^{*}f_{0}\|\frac{\|K^{*}f_{0}\|}{A}\bigg{)}$
$\displaystyle=$ $\displaystyle-\frac{\|K^{*}f_{0}\|^{2}}{2A}.$

In view of (1.2), for every $v\in\cal{H}$ we have

$\displaystyle J_{f_{0},\sigma_{S}}(v)=\bigg{(}\frac{1}{2}\langle
Sv,v\rangle-\langle KK^{*}f_{0},v\rangle\bigg{)}$ $\displaystyle\leq$
$\displaystyle\bigg{(}\frac{B}{2}\|v\|^{2}-\langle
KK^{*}f_{0},v\rangle\bigg{)}.$

Thus

$\displaystyle\min_{v\in{\cal H}}J_{f_{0},\sigma_{S}}(v)$ $\displaystyle\leq$
$\displaystyle\min_{v\in{\cal H}}\bigg{(}\frac{B}{2}\|v\|^{2}-\langle
KK^{*}f_{0},v\rangle\bigg{)}$ $\displaystyle\leq$
$\displaystyle\frac{B\alpha^{2}}{2}\|f_{0}\|^{2}-\alpha\|K^{*}f_{0}\|^{2}$
$\displaystyle<$ $\displaystyle 0,$

whenever

$\displaystyle 0<\alpha<\frac{2}{B}\frac{\|K^{*}f_{0}\|^{2}}{\|f_{0}\|^{2}}.$

The minimum of the function
$\alpha\mapsto\frac{B\alpha^{2}}{2}\|f_{0}\|^{2}-\alpha\|K^{*}f_{0}\|^{2}$ is
attained in point $\alpha=\frac{\|K^{*}f_{0}\|^{2}}{4B\|f_{0}\|^{2}}$.
Therefore,

$\displaystyle\frac{B}{2}\,\alpha^{2}\|f_{0}\|^{2}-\alpha\|K^{*}f_{0}\|^{2}$
$\displaystyle=$
$\displaystyle\frac{B}{2}\frac{\|K^{*}f_{0}\|^{4}}{16B^{2}\|f_{0}\|^{4}}\|f_{0}\|^{2}-\frac{\|K^{*}f_{0}\|^{2}}{4B\|f_{0}\|^{2}}\|K^{*}f_{0}\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\|K^{*}f_{0}\|^{2}}{16B\|f_{0}\|^{2}}-\frac{\|K^{*}f_{0}\|^{4}}{4B\|f_{0}\|^{2}}$
$\displaystyle=$
$\displaystyle-\frac{7}{32}\frac{\|K^{*}f_{0}\|^{4}}{B\|f_{0}\|^{2}},$

as claimed. ∎

##  3 Construction of Scalable $K$-Frames

Throughout this section, $H$ denotes a separable infinite dimensional Hilbert
space. First, we give a slight generalization of the concept of scalable
frames for $K$-frames; in fact, a $K$-frame
$\\{f_{j}\\}_{j\in\mathbb{Z}}\subseteq H$ is called a $K_{s}$-frame with
scalling $\\{c_{j}\\}\in\ell^{\infty}(J)$, briefly $K_{s}$-frame and denoted
by $(f_{j},c_{j})_{j\in\mathbb{Z}}$, if $\\{c_{j}f_{j}\\}_{j\in\mathbb{Z}}$ is
a _Parseval $K$-frame_, i.e., for every $f\in H$, we have

$\displaystyle\sum_{j\in\mathbb{Z}}\langle
f,c_{j}f_{j}\rangle\,c_{j}f_{j}=KK^{*}f;$

or equivalently,

$\sum_{j\in\mathbb{Z}}|\langle f,c_{j}f_{j}\rangle|^{2}=\|K^{*}f\|^{2}.$

######  Theorem 3.1.

Let $U,V\in\mathcal{L}(H)$ and $\\{f_{j}\\}_{j\in\mathbb{Z}}$ be a sequence in
$H$. Then the following statements hold.

_(i)_ If $\\{f_{j}\\}_{j\in\mathbb{Z}}$ is a scalable frame, then
$\\{Uf_{j}\\}_{j\in\mathbb{Z}}$ is a $U_{s}$-frame with the same scalling of
$\\{f_{j}\\}_{j\in\mathbb{Z}}$.

_(ii)_ If $\\{f_{j}\\}_{j\in\mathbb{Z}}$ is a $U_{s}$-frame, then sequence
$\\{Vf_{j}\\}_{j\in\mathbb{Z}}$ is a $VU_{s}$-frame with the same scalling of
$\\{f_{j}\\}_{j\in\mathbb{Z}}$.

_(iii)_ If $\\{f_{j}\\}_{j\in\mathbb{Z}}$ and $\\{Uf_{j}\\}_{j\in\mathbb{Z}}$
are $K_{s}$-frames with the same scallings, then
$\\{f_{j}\\}_{j\in\mathbb{Z}}$ is a $UK_{s}$-frame with the same scalling of
$\\{f_{j}\\}_{j\in\mathbb{Z}}$.

_(iv)_ If $\\{f_{j}\\}_{j\in\mathbb{Z}}$ is a $K_{s}$-frame and $N\geq 1$,
then $\\{K^{N}f_{j}\\}_{j\in\mathbb{Z}}$ is a $K_{s}^{N+1}$-frame with the
same scalling of $\\{f_{j}\\}_{j\in\mathbb{Z}}$.

###### Proof.

Let $T\in\mathcal{L}(H)$ and $\\{c_{j}\\}$ be a sequence in
$\ell^{\infty}(J)$. Then for every $f\in H$, we have

$\displaystyle\sum_{j\in\mathbb{Z}}\langle f,c_{j}Tf_{j}\rangle\,c_{j}Tf_{j}$
$\displaystyle=$ $\displaystyle T\bigg{(}\sum_{j\in\mathbb{Z}}\langle
f,c_{j}Tf_{j}\rangle\,c_{j}f_{j}\bigg{)}$ $\displaystyle=$ $\displaystyle
T\bigg{(}\sum_{j\in\mathbb{Z}}\langle
T^{*}f,c_{j}f_{j}\rangle\,c_{j}f_{j}\bigg{)}.$

This proves (i) and (ii) if we set $T=U$ and $T=V$, respectively.

Let $(f_{j},c_{j})_{j\in\mathbb{Z}}$ and $(Uf_{j},c_{j})_{j\in\mathbb{Z}}$ be
$K_{s}$-frames. Then

$\displaystyle(UK)(UK)^{*}f$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z}}\langle f,c_{j}Uf_{j}\rangle\,c_{j}Uf_{j}$
$\displaystyle=$ $\displaystyle KK^{*}f$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z}}\langle f,c_{j}f_{j}\rangle\,c_{j}f_{j}.$

So, $(f_{j},c_{j})_{j\in\mathbb{Z}}$ is $UK_{s}$-frame. Thus (iii) holds.

Let $(f_{j},c_{j})_{j\in\mathbb{Z}}$ be a $K_{s}$-frame. Then for every $f\in
H$, we have

$\displaystyle\sum_{j\in\mathbb{Z}}|\langle f,c_{j}K^{N}f_{j}\rangle|^{2}$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z}}|\langle(K^{N})^{*}f,c_{j}f_{j}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\|K^{*}({K^{N}})^{*}f\|^{2}$ $\displaystyle=$
$\displaystyle\|(K^{N+1})^{*}f)\|^{2}.$

Therefore, $(K^{N}f_{j},c_{j})_{j\in\mathbb{Z}}$ is $K^{N+1}_{s}$-frame. That
is, (iv) is true. ∎

######  Theorem 3.2.

Let $T\in\mathcal{L}(H)$ and $(f_{j},c_{j})_{j\in\mathbb{Z}}$ be a
$K_{s}$-frame. If $TK=KT$ and $T^{*}$ is isometry, then
$(Tf_{j},c_{j})_{j\in\mathbb{Z}}$ is a $K_{s}$-frame.

###### Proof.

Let $(f_{j},c_{j})_{j\in\mathbb{Z}}$ be a $K_{s}$-frame. Then for every $f\in
H$, we have

$\displaystyle\sum_{j\in\mathbb{Z}}|\langle f,c_{j}Tf_{j}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\sum_{j\in\mathbb{Z}}|\langle
T^{*}f,c_{j}f_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\|K^{*}T^{*}f\|^{2}$ $\displaystyle=$
$\displaystyle\|(TK)^{*}f\|^{2}$ $\displaystyle=$
$\displaystyle\|(KT)^{*}f\|^{2}$ $\displaystyle=$
$\displaystyle\|T^{*}(K^{*}f)\|^{2}$ $\displaystyle=$
$\displaystyle\|K^{*}f\|^{2}.$

Thus $(Tf_{j},c_{j})_{j\in\mathbb{Z}}$ is a $K_{s}$-frame. ∎

######  Theorem 3.3.

Let $T\in\mathcal{L}(H)$ be an invertible operator,
$\\{f_{j}\\}_{j\in\mathbb{Z}}$ be a $K$-frame for $H$ and
$\\{c_{j}\\}\in\ell^{\infty}(J)$. Then $(Tf_{j},c_{j})_{j\in\mathbb{Z}}$ is a
$K_{s}$-frame if and only if the frame operator for
$\\{c_{j}Tf_{j}\\}_{j\in\mathbb{Z}}$ is $(T^{-1}K)(T^{-1}K)^{*}$.

###### Proof.

Let $S$ be the operator frame for $\\{c_{j}f_{j}\\}_{j\in\mathbb{Z}}$. Then
for every $f\in H$, we have

$\displaystyle\sum_{j\in\mathbb{Z}}\langle f,c_{j}Tf_{j}\rangle\,c_{j}Tf_{j}$
$\displaystyle=$ $\displaystyle T\bigg{(}\sum_{j\in\mathbb{Z}}\langle
T^{*}f,c_{j}f_{j}\rangle\,c_{j}f_{j}\bigg{)}$ (3.1) $\displaystyle=$
$\displaystyle TST^{*}f.$

On the other hand, $(Tf_{j},c_{j})_{j\in\mathbb{Z}}$ is a $K_{s}$-frame. Thus

$\displaystyle\sum_{j\in\mathbb{Z}}\langle
f,c_{j}Tf_{j}\rangle\,c_{j}Tf_{j}=KK^{*}f.$ (3.2)

By using (3.1) and (3.2), $(Tf_{j},c_{j})_{j\in\mathbb{Z}}$ is a $K_{s}$-frame
if and only if $S=(T^{-1}K)(T^{-1}K)^{*}$. ∎

##  4 Piecewise Scalable $K$-Frames

A $K$-frame $\\{f_{j}\\}_{j=1}^{n}$ for the $n$-dimensional Hilbert space
$\mathbb{R}^{n}$ is called a piecewise scalable $K$-frame (briefly,
$K_{s}^{p}$-frame) if there exist an orthogonal projection
$P:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and constants
$\\{a_{j},b_{j}\\}_{j=1}^{n}$ such that for every $f\in\mathbb{R}^{n}$

$\displaystyle\sum_{j=1}^{n}|\langle
f,a_{j}Pf_{j}+b_{j}(I-P)f_{j}\rangle|^{2}=\|K^{*}f\|^{2}.$

We denote this notion by $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$. In the following,
let $P$ be an orthogonal projection on ${\mathbb{R}}^{n}$ and we set

$\displaystyle X:=P(\mathbb{R}^{n})\quad\hbox{and}\quad
Y:=(I-P)(\mathbb{R}^{n}).$

One can prove that if $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame
for $\mathbb{R}^{n}$, then $(Pf_{j},a_{j})_{j=1}^{n}$ and
$((I-P)f_{j},b_{j})_{j=1}^{n}$ are $K_{s}$-frames for $X$ and $Y$,
respectively. To prove other our results, we need the following lemma.

######  Lemma 4.1.

Let $\\{f_{j}\\}_{j=1}^{n}$ be a $K$-frame for $\mathbb{R}^{n}$ and $KP=PK$.
If $\\{f_{j}\\}_{j=1}^{n}$ is a Parseval $K$-frame for $\mathbb{R}^{n}$, then
$\\{Pf_{j}\\}_{j=1}^{n}$ is a Parseval $K$-frame for $X$.

###### Proof.

For every $f\in X$, we have

$\displaystyle\sum_{j=1}^{n}|\langle f,Pf_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}|\langle Pf,f_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\|K^{*}Pf\|^{2}$ $\displaystyle=$ $\displaystyle\|PK^{*}f\|^{2}$
$\displaystyle=$ $\displaystyle\|K^{*}f\|^{2}.$

Also, note that if $PK=KP$, then $\hbox{ran}(K^{*})\subseteq\hbox{ran}(P)$. ∎

######  Theorem 4.2.

Let $\\{f_{j}\\}_{j=1}^{n}$ be a $K$-frame for $\mathbb{R}^{n}$ and $KP=PK$.
Then $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame if and only if

_(i)_ Both $(Pf_{j},a_{j})_{j=1}^{n}$ and $((I-P)f_{j},b_{j})_{j=1}^{n}$ are
$K_{s}$-frame for $X$ and $Y$, respectively.

_(ii)_ For every $f\in\mathbb{R}^{n}$, we have

$\displaystyle\sum_{j=1}^{n}a_{j}b_{j}\langle f,Pf_{j}\rangle\langle
f,(I-P)f_{j}\rangle=0.$

###### Proof.

First, note that if (i) holds, then for every $f\in H$, we have

$\displaystyle\sum_{j=1}^{n}|\langle
f,a_{j}Pf_{j}+b_{j}(I-P)f_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}|\langle
f,a_{j}Pf_{j}\rangle|^{2}+\sum_{j=1}^{n}|\langle
f,b_{j}(I-P)f_{j}\rangle|^{2}$ $\displaystyle+$ $\displaystyle
2\sum_{j=1}^{n}a_{j}b_{j}\langle f,Pf_{j}\rangle\langle f,(I-P)f_{j}\rangle$
$\displaystyle=$ $\displaystyle\|PK^{*}f\|+\|(I-P)K^{*}f\|^{2}$
$\displaystyle+$ $\displaystyle 2\sum_{j=1}^{n}a_{j}b_{j}\langle
f,Pf_{j}\rangle\langle f,(I-P)f_{j}\rangle$ $\displaystyle=$
$\displaystyle\|K^{*}f\|^{2}+2\sum_{j=1}^{n}a_{j}b_{j}\langle
f,Pf_{j}\rangle\langle f,(I-P)f_{j}\rangle.$

Assume now that $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame. From
Lemma 4.1 and the fact

$\displaystyle P(a_{j}Pf_{j}+b_{j}(I-P)f_{j})=a_{j}Pf_{j},$

we infer that $\\{a_{j}Pf_{j}\\}_{j=1}^{n}$ is a Parseval $K$-frame for $X$.
Therefore, $\\{b_{j}(I-P)f_{j}\\}_{j=1}^{n}$ is a Parseval $K$-frame for $Y$.
That is, (i) is proved. The converse follows at once from (4). ∎

######  Corollary 4.3.

Let $PK=KP$ and two of the following three statements be true. Then the third
statement is also true.

_(i)_ $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame.

_(ii)_ Both $(Pf_{j},a_{j})_{j=1}^{n}$ and $((I-P)f_{j},b_{j})_{j=1}^{n}$ are
$K_{s}$-frame for $X$ and $Y$, respectively.

_(iii)_ For every $f\in\mathbb{R}^{n}$, we have

$\displaystyle\sum_{j=1}^{n}a_{j}b_{j}\langle f,Pf_{j}\rangle\langle
f,(I-P)f_{j}\rangle=0.$

In the next result, we let $T_{1}$ and $T_{2}$ be analysis operators of
$\\{Pf_{j}\\}_{j=1}^{n}$ and $\\{(I-P)f_{j}\\}_{j=1}^{n}$, respectively.

######  Corollary 4.4.

Let $\\{f_{j}\\}_{j=1}^{n}$ be a $K$-frame for $\mathbb{R}^{n}$ and $PK=KP$.
If there exists a proper non-empty subset $I$ of $\\{1,...,n\\}$ such that
$(Pf_{j},c_{j})_{j\in I}$ and $((I-P)f_{j},d_{j})_{j\in I^{c}}$ are
$K_{s}$-frames for $X$ and $Y$, respectively, then there exist constants
$\\{a_{j}\\}_{j=1}^{n}$ and $\\{b_{j}\\}_{j=1}^{n}$ such that
$(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame for $\mathbb{R}^{n}$.

###### Proof.

Let $I$ be a proper non-empty subset of $\\{1,...,n\\}$ such that
$(Pf_{j},c_{j})_{j\in I}$ and $((I-P)f_{j},d_{j})_{j\in I^{c}}$ are
$K_{s}$-frames for $X$ and $Y$, respectively. Set

$\displaystyle a_{j}=\left\\{\begin{array}[]{ll}c_{j},&\hbox{$j\in I$;}\\\
0,&\hbox{$j\in I^{c}$}\end{array}\right.\quad\hbox{and}\quad\quad
b_{j}=\left\\{\begin{array}[]{ll}0,&\hbox{$j\in I$;}\\\ d_{j},&\hbox{$j\in
I^{c}$.}\end{array}\right.$

Then for every $f\in X$, we have

$\displaystyle\sum_{i=1}^{n}|\langle f,a_{j}Pf_{j}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\sum_{i\in I}|\langle
f,a_{j}Pf_{j}\rangle|^{2}+\sum_{i\in I^{c}}|\langle f,a_{j}Pf_{j}\rangle|^{2}$
$\displaystyle=$ $\displaystyle\|K^{*}f\|^{2}.$

So $(Pf_{j},a_{j})_{j\in I}$ is a $K_{s}$-frame for $X$. Similarly, one can
prove that $((I-P)f_{j},b_{j})_{j\in I^{c}}$ is a $K_{s}$-frame for $Y$. But,
$a_{j}b_{j}=0$ for all $j=1,...,n$. Then by Theorem 4.2,
$(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame for $\mathbb{R}^{n}$.
∎

As an immediate consequence of Theorem 4.2 and Corollary 4.4, we have the
following result.

######  Corollary 4.5.

Let $\\{f_{j}\\}_{j=1}^{n}$ be a $K$-frame for $\mathbb{R}^{n}$ and $PK=KP$.
If there exists a non-empty proper subset $I$ of $\\{1,...,n\\}$ such that
$(Pf_{j},c_{j})_{j\in I}$ and $((I-P)f_{j},d_{j})_{j\in I^{c}}$ are
$K_{s}$-frames for $X$ and $Y$, respectively, then $(Pf_{j},c_{j})_{j=1}^{n}$
and $((I-P)f_{j},d_{j})_{j=1}^{n}$ are $K_{s}$-frames for $X$ and $Y$,
respectively.

######  Proposition 4.6.

Let $\\{f_{j}\\}_{j=1}^{n}$ be a $K$-frame for $\mathbb{R}^{n}$ and $PK=KP$.
Then the following assertions are equivalent.

_(a)_ $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}$-frame.

_(b)_ If $D_{1}$ and $D_{2}$ are diagonal operators on $\ell^{2}(n)$ with
diagonal elements $\\{a_{j}\\}_{j=1}^{n}$ and $\\{b_{j}\\}_{j=1}^{n}$,
respectively, then

$\displaystyle T_{1}^{*}D_{1}^{2}T_{1}=KK^{*}_{\mkern 1.0mu\vrule
height=5.42499pt\mkern 2.0muX},~{}~{}T_{2}^{*}D_{2}^{2}T_{2}=KK^{*}_{\mkern
1.0mu\vrule height=5.42499pt\mkern
2.0muY}~{}~{}\emph{and}~{}~{}T_{1}^{*}D_{1}D_{2}T_{2}=0.$

###### Proof.

Let $\\{f_{j}\\}_{j=1}^{n}$ be a $K^{p}_{s}$-frame for $\mathbb{R}^{n}$. By
Theorem 4.2 (i), $\\{a_{j}Pf_{j}\\}_{j=1}^{n}$ and
$\\{b_{j}(I-P)f_{j}\\}_{j=1}^{n}$ are $K_{s}$-frame for $X$ and $Y$,
respectively. It follows that

$T_{1}^{*}D_{1}^{2}T_{1}=S_{1}=KK^{*}_{\mkern 1.0mu\vrule
height=5.42499pt\mkern 2.0muX}\quad\quad\hbox{and}\quad\quad
T_{2}^{*}D_{2}^{2}T_{2}=S_{2}=KK^{*}_{\mkern 1.0mu\vrule
height=5.42499pt\mkern 2.0muY},$

where $S_{1}$ and $S_{2}$ are the frame operators of
$\\{a_{j}Pf_{j}\\}_{j=1}^{n}$ and $\\{b_{j}(I-P)f_{j}\\}_{j=1}^{n}$,
respectively. On the other hand,

$\displaystyle\langle
T_{1}^{*}D_{1}D_{2}T_{2}(f),f\rangle=\bigg{\langle}\sum_{j=1}^{n}a_{j}b_{j}\langle
f,Pf_{j}\rangle\langle f,(I-P)f_{j}\rangle,f\bigg{\rangle}$ (4.3)

for every $f\in\mathbb{R}^{n}$. Another application of Theorem 4.2(ii), shows
that

$\displaystyle\langle T_{1}^{*}D_{1}D_{2}T_{2}(f),f\rangle=0$

for every $f\in\mathbb{R}^{n}$. So (a)$\Rightarrow$(b).

To complete the proof, note that if $T_{1}^{*}D_{1}^{2}T_{1}=KK^{*}_{\mkern
1.0mu\vrule height=5.42499pt\mkern 2.0muX}$ and
$T_{2}^{*}D_{2}^{2}T_{2}=KK^{*}_{\mkern 1.0mu\vrule height=5.42499pt\mkern
2.0muY}$, then $\\{a_{j}Pf_{j}\\}_{j=1}^{n}$ and
$\\{b_{j}(I-P)f_{j}\\}_{j=1}^{n}$ are $K_{s}$-frame for $X$ and $Y$,
respectively. Also, if $T_{1}^{*}D_{1}D_{2}T_{2}=0$, then by (4.3), we have

$\displaystyle\sum_{j=1}^{n}a_{j}b_{j}\langle f,Pf_{j}\rangle\langle
f,(I-P)f_{j}\rangle=0.$

Hence (b)$\Rightarrow$(a). ∎

######  Theorem 4.7.

Let $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ be a $K^{p}_{s}$-frame. Let
$Q:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be an orthogonal projection and
$U:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a unitary operator such that

$\displaystyle UP=QU\quad\emph{and}\quad UK=KU.$

Then $(Uf_{j},Q,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}-$frame.

###### Proof.

Let $(f_{j},P,a_{j},b_{j})_{j=1}^{n}$ be a $K^{p}_{s}$-frame. Then for every
$f\in\mathbb{R}^{n}$, we obtain

$\displaystyle\sum_{j=1}^{n}|\langle
f,a_{j}Q\,Uf_{j}+b_{j}(I-Q)\,Uf_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}|\langle
f,a_{j}U\,Pf_{j}+b_{j}U\,(I-P)f_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}|\langle
U^{*}f,a_{j}Pf_{j}+b_{j}(I-P)f_{j}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\|K^{*}\,U^{*}f\|^{2}$ $\displaystyle=$
$\displaystyle\|U^{*}\,K^{*}f\|^{2}$ $\displaystyle=$ $\displaystyle\langle
U\,U^{*}K^{*}f,K^{*}f\rangle$ $\displaystyle=$ $\displaystyle\|K^{*}f\|^{2}.$

So, $(Uf_{j},Q,a_{j},b_{j})_{j=1}^{n}$ is a $K^{p}_{s}-$frame. ∎

##  5 Compliance with ethical standards

Conflict of interest: All authors declare that they have no conflict of
interest.
Data Availability Statement: No data sets were generated or analyzed during
the current study.

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