An Information-Theoretic Approach to Diversity Evaluation of Prompt-based Generative Models

## Abstract

Abstract Text-conditioned generation models are commonly evaluated based on the quality of the generated data and its alignment with the input text prompt. On the other hand, several applications of prompt-based generative models require sufficient diversity in the generated data to ensure the models’ capability of generating image and video samples possessing a variety of features. However, most existing diversity metrics are designed for unconditional generative models, and thus cannot distinguish the diversity arising from variations in text prompts and that contributed by the generative model itself. In this work, our goal is to quantify the prompt-induced and model-induced diversity in samples generated by prompt-based models. We propose an information-theoretic approach for internal diversity quantification, where we decompose the kernel-based entropy H ⁢ ( X ) 𝐻 𝑋 H(X) italic_H ( italic_X ) of the generated data X 𝑋 X italic_X into the sum of the conditional entropy H ⁢ ( X | T ) 𝐻 conditional 𝑋 𝑇 H(X|T) italic_H ( italic_X | italic_T ) , given text variable T 𝑇 T italic_T , and the mutual information I ⁢ ( X ; T ) 𝐼 𝑋 𝑇 I(X;T) italic_I ( italic_X ; italic_T ) between the text and data variables. We introduce the Conditional-Vendi score based on H ⁢ ( X | T ) 𝐻 conditional 𝑋 𝑇 H(X|T) italic_H ( italic_X | italic_T ) to quantify the internal diversity of the model and the Information-Vendi score based on I ⁢ ( X ; T ) 𝐼 𝑋 𝑇 I(X;T) italic_I ( italic_X ; italic_T ) to measure the statistical relevance between the generated data and text prompts. We provide theoretical results to statistically interpret these scores and relate them to the unconditional Vendi score. We conduct several numerical experiments to show the correlation between the Conditional-Vendi score and the internal diversity of text-conditioned generative models.
The codebase is available at https://github.com/mjalali/conditional-vendi .

## 1 Introduction

Prompt-based generative models, including text-to-image and text-to-video generation schemes, are widely used in various artificial intelligence (AI) applications. In prompt-based generative AI, the sample creation process begins with a text input and produces a random output aligned with that text. The conditional nature of this sample generation distinguishes prompt-based generative models from standard unconditional generative models where the objective is to produce samples distributed similarly to real data without any guiding input prompt. Since most evaluation metrics for generative models had been developed for unconditional models in the previous decade, the recent literature has sought to create scores tailored specifically for text-conditioned generative models.

The existing evaluation metrics for prompt-based generative models typically focus on fidelity and relevance in sample generation, i.e., they assess the visual quality of the produced samples and their alignment with the input prompt. Relevance is often measured by calculating a similarity score between a shared embedding of the text and image samples, e.g. in ClipScore [] which utilizes the CLIP embeddings of text and image data. Such shared embedding-based evaluation mechanisms have been further adapted to quantify the aesthetics, semantic consistency, and compositional accuracy of the generated data based on the input text prompt.

On the other hand, the diversity performance of prompt-based generative models has not been exclusively studied in the literature. The diversity scores proposed for unconditional generative models, such as Recall [, ], Coverage [], Vendi [, ], and RKE [], are often applied to quantify the variety of generated samples. However, in text-based generative models, the generated samples are typically produced in response to different input prompts, where the variation in input texts can significantly contribute to the diversity of the generated image or video samples. Thus, the diversity of data produced by prompt-based models is influenced by two main factors: 1) the variety of input prompts, and 2) the internal diversity of the model introducing randomness into the output samples. Figure 1 illustrates examples of prompt-induced diversity, where the variety of generated images is mostly due to the different prompts and the generated data has little variety in other details (images of similar people with similar poses), and model-induced diversity where the details not specified by the prompts vary significantly between generated images. This decomposition of diversity in text-based generative models has not been studied in the existing literature on conditional generative models including text-based image and video generation.

In this work, we focus on quantifying the two diversity components mentioned for prompt-based generative models. To this end, we propose an information-theoretic decomposition of the diversity of the model’s output data, $X$. The proposed decomposition is based on a classical identity in information theory, which shows that for variables $X$ and $T$, the Shannon entropy $H(X)$, representing the uncertainty of $X$, can be decomposed into two terms as follows:

$$ $H(X)\;=\;H\bigl{(}X|T\bigr{)}+I\bigl{(}X;T\bigr{)}$ $$

Here, $H(X|T)$ denotes the conditional entropy of data $X$ given the text variable $T$, which we interpret as the internal diversity of the text-based generative model not caused by variation in the input text $T$. Furthermore, the mutual information term $I(X;T)$ can be viewed as a measure of statistical relevance between the text $T$ and the generated data $X$, quantifying how much information the model’s output conveys about the input text.

Figure: Figure 1: Illustration of *prompt-induced diversity*, where the diversity of generated images follows the variety of prompts and has little variation in other details, vs. *model-induced diversity*, where the diversity of images for similar prompts is due to the generation model.
Refer to caption: x1.png

To mathematically define the entropy-based scores, we follow the kernel-matrix-based entropy definition, which has been applied by [, , ] to unconditional generative models. These references apply the matrix-based entropy in quantum information theory, that is the entropy of the eigenvalues of the kernel matrix of generated data $X$, to measure the diversity of an unconditional model’s generated data. To extend the framework to conditional prompt-based generative models, we utilize the definition of matrix-based conditional entropy proposed by []. This work provides a definition for the conditional entropy of two general positive semi-definite matrices, which we select to be the kernel matrices of generated data $X$ and text $T$. Following these definitions, our work extends the entropy-based approach in [, ] to conditional generative models. We define the *Conditional-Vendi* and *Information-Vendi* scores which decompose the unconditional Vendi score to model-induced and prompt-induced diversity measures.

To statistically interpret the defined scores, we first derive the statistic estimated by these scores from empirically generated samples. We show how the target statistic can be formulated in terms of the kernel covariance matrix of the Kronecker product of text $T$ and data $X$ vectors. Also, to provide an operational meaning for the proposed score, we prove that given a mixture distribution $P_{T}$ for text data with a hidden group variable $G\in\{1,\ldots,m\}$, the defined conditional entropy approximates the expectation of the entropy $H(X|G=g)$ given prompt type $g\sim P_{G}$. This result connects our proposed diversity assessment approach to the Vendi [, ] and RKE [] scores, showing that the Conditional-Vendi score is an aggregation of the unconditional Vendi scores for different prompt groups.

We numerically evaluate the proposed diversity scores for standard text-to-image, text-to-video, and image-captioning generative models. In our experiments, we simulate text-based generative models for which the ground-truth rankings of internal diversity and relevance are known. Our experimental results validate the consistency of our proposed information-theoretic scores and the ground-truth ranking of the models. We further decompose the Conditional-Vendi score across different modes of input text data, evaluating the models’ internal diversity across different types of input text. The following is a summary of the contributions of this work:

- •
Highlighting the diversity evaluation task in the context of conditional and prompt-based generative models,
- •
Proposing an information-theoretic framework for decomposing the diversity of generated data into prompt-induced and model-induces components to evaluate the internal diversity of prompt-based generative models
- •
Providing statistical analysis of the proposed scores and interpreting them as the conditional expectation of entropy scores given the text prompt
- •
Presenting numerical results on the consistency between the conditional entropy score and the model-induced diversity of text-to-image and text-to-video generative models.

## 2 Related Work

Evaluation of deep generative models:
The existing metrics for the evaluation of generative models can be divided into reference-dependent and reference-free categories, as discussed in []. As one type of reference-dependent metrics, a distance between generated and reference distributions is measured using metrics such as FID [] and KID []. Other reference-based metrics such as the Inception Score [], GAN-train/GAN-test [], Precision/Recall [, ], and Density/Coverage [] are defined to quantify the diversity and quality of generated data in comparison to the samples in the real dataset.
In addition, assessing memorization and novelty has been studied in several references, including the authenticity score [] and Feature Likelihood Divergence [] to assess generalizability, and the rarity score [], KEN [] and FINC [] proposed to assess novelty. Note that the memorization metrics are inherently reference-based. In contrast, reference-free evaluations [, ] measure diversity [] and quality based only based on the generated data. Specifically, the Vendi [, ] and RKE scores [] fall into this category.

Evaluation of conditional generative models:
The evaluation of prompt-based generative models, including text-to-image and text-to-video models, has been studied in several related works. Most of the existing evaluation metrics attempt to measure the correlation between the prompt and the output. A standard metric for measuring the alignment of prompt and image is CLIPScore [], which measures the cosine similarity of the paired data using CLIP embedding. Some other works introduce different benchmarks and sets of prompts to evaluate different aspects. HEIM [] assesses twelve aspects of the sample generation, including text-image alignment, image quality, and bias. As noted by [], standard metrics focusing on style, aesthetics, and image quality, may overlook the diversity of images given a particular prompt. In their work, they measure diversity separately for each prompt using a similarity function and average the scores for the prompts. [] follows a similar approach to the Vendi score. We note that both these methods require generating multiple images per prompt with different seeds to measure the score. On the other hand, our proposed Conditional-Vendi score does not require multiple sample generations per prompt and instead analyzes the types of text prompts in the assessment. Our theoretical results interpret Conditional-Vendi as an aggregation of the scores over text types.

Information measures for evaluating conditional generative models:
[] utilize the mutual information (MI) between continuous text and image variables, and propose the Mutual Information Divergence (MID) score. This work fits a multivariate Gaussian distribution to the text and image data and then estimates their mutual information to quantify a relevance metric for conditional generative models. We note that our proposed $\mathrm{Information}\text{-}\mathrm{Vendi}$ score is based on the matrix-based entropy score by [] which is different from the MI between Gaussian vectors fitted to the text and image data used in the MID score. Different from MID, $\mathrm{Information}\text{-}\mathrm{Vendi}$ relies on kernel similarity values to identify a cluster variable for MI calculation.

## 3 Preliminaries

Throughout the work, we focus on a conditional generative model that produces a data vector $X\in\mathcal{X}$ given an input text prompt $T\in\mathcal{T}$ according to a conditional distribution $P_{X|T}$, i.e., for text prompt $T=t$ the model outputs a random sample following $P_{X|T=t}$. We consider $n$ sample pairs $(t_{i},x_{i})\sim P_{T}\times P_{X|T}$ where each text prompt $t_{i}$ is drawn independently from the distribution $P_{T}$ and then the generated sample $x_{i}$ is generated according to $P_{X|T=t_{i}}$. Our goal is to quantify the internal diversity of the prompt-based generative model, influencing the variety of data generated $x_{1},\ldots,x_{n}$ independently of the diversity of input texts $t_{1},\ldots,t_{n}$.

In this section, we first discuss existing entropy-based approaches to the diversity evaluation of unconditional generative models and then review the definition of matrix-based conditional entropy and mutual information in the literature.

### 3.1 Entropy-based diversity scores for unconditional generative models

Consider generated samples $x_{1},\ldots,x_{n}\in\mathcal{X}$ following the distribution $P_{X}$ of an unconditional generative model. For a kernel function $k:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$, the kernel similarity matrix $K\in\mathbb{R}^{n\times n}$ is:

$$ $K=\begin{bmatrix}k(x_{1},x_{1})&\cdots&k(x_{1},x_{n})\\ \vdots&\ddots&\vdots\\ k(x_{n},x_{1})&\cdots&k(x_{n},x_{n})\end{bmatrix}$ (1) $$

Following the standard definition, a kernel function $k$ satisfies the positive semidefinite property (PSD), which means that the above kernel matrix will be PSD for any arbitrary selection of data points $x_{1},\ldots,x_{n}\in\mathcal{X}$, i.e., all its eigenvalues are non-negative. A well-known example of kernel functions is the Gaussian (RBF) kernel, which for a bandwdith parameter $\sigma$ is defined as:

$$ $k(x,x^{\prime})\,=\,\exp\Bigl{(}\frac{-\bigl{\|}x-x^{\prime}\bigr{\|}^{2}_{2}} {2\sigma^{2}}\Bigr{)}$ (2) $$

Assuming that a kernel function $k$ is normalized, i.e. $k(x,x)=1$ for every $x\in\mathbb{R}^{d}$, then the non-negative eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ of $\frac{1}{n}K$ will add up to $1$, implying that they represent a probability model. In the literature, [, , ] propose using the general order-$\alpha$ Renyi entropy of the probability model as the model’s diversity score, defined as follows for $\frac{1}{n}K$ in equation 1:

$$ $H_{\alpha}\bigl{(}X\bigr{)}\,:=\,H_{\alpha}\bigl{(}\frac{1}{n}K\bigr{)}\,=\, \frac{1}{1-\alpha}\log\Bigl{(}\sum_{i=1}^{n}\lambda_{i}^{\alpha}\Bigr{)}$ (3) $$

In the special case of $\alpha=1$, the above definition results in the Shannon-entropy of eigenvalues $H_{1}(\frac{1}{n}K)=\sum_{i=1}^{n}\lambda_{i}\log(1/\lambda_{i})$. Also, we note that the Vendi and RKE scores defined by [, ] are the exponential of the defined entropy measure, where

$$ $\mathrm{Vendi}_{\alpha}\bigl{(}x_{1},\ldots,x_{n}\bigr{)}:=\exp\Bigl{(}H_{ \alpha}\bigl{(}\frac{1}{n}K\bigr{)}\Bigr{)}$ (4) $$

To statistically interpret the entropy measures of the samples’ kernel matrix, [, ] note that the normalized kernel matrix $\frac{1}{n}K$ shares the same non-zero eigenvalues with the empirical kernel covariance matrix $\widehat{C}_{X}$ defined as:

$$ $\widehat{C}_{X}\,=\,\frac{1}{n}\sum_{i=1}^{n}\phi(X_{i})\phi(X_{i})^{\top}$ (5) $$

Here, $\phi:\mathcal{X}\rightarrow\mathbb{R}^{d}$ denotes the kernel feature map satisfying the relation $k(x,x^{\prime})=\langle\phi(x),\phi(x^{\prime})\rangle$ for every $x,x^{\prime}\in\mathcal{X}$ where $\langle\cdot,\cdot\rangle$ is the standard inner-product in the $\mathbb{R}^{d}$ space. As a result, the entropy of the kernel matrix $K$’s eigenvalues equals the entropy of $\widehat{C}_{X}$. Note that $\widehat{C}_{X}$ is the empirical estimation of the underlying kernel covariance matrix $C_{X}=\mathbb{E}_{X\sim P_{X}}\bigl{[}\phi(X)\phi(X)^{\top}\bigr{]}$.

### 3.2 Matrix-based Conditional Entropy and Mutual Information

In the previous subsection, we reviewed the standard definition of order-$\alpha$ matrix-based entropy for PSD matrices. Here, we discuss an extension proposed by [] to define matrix-based conditional entropy and mutual information for two variables $X\in\mathcal{X}$ and $T\in\mathcal{T}$. For variables $X$ and $T$, we consider normalized kernel functions $k_{X}:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ and $k_{T}:\mathcal{T}\times\mathcal{T}\rightarrow\mathbb{R}$, where the kernel functions satisfy $k_{X}(x,x)=1$ and $k_{T}(t,t)=1$ for every input $x$ and $t$. Given the two kernel functions, [] define the order-$\alpha$ matrix-based joint entropy $H_{\alpha}(X,T)$ as the order-$\alpha$ entropy of the PSD matrix $\frac{1}{n}K_{X}\odot K_{T}$, in which $K_{X}$ and $K_{T}$ are the kernel matrices of $X$ and $T$ samples and $\odot$ denotes the entry-wise Hadamard product.

Note that the Hadamard product $K_{X}\odot K_{T}$ represents the kernel matrix of concatenated samples $[x_{i},t_{i}]$, where we consider the kernel function $k_{X,T}([x,t],[x^{\prime},t^{\prime}])=k_{X}(x,x^{\prime})k_{T}(t,t^{\prime})$ to be the product of marginal kernel functions. This definition is sensible, since the joint similarity value, taking value over $[0,1]$ in Gaussian kernels, is considered to be the multiplication of the similarity scores for the text and output data vectors. Then, [] propose defining conditional entropy $H_{\alpha}(X|T)$ as the difference between the joint and marginal entropy values:

$$ $H_{\alpha}\bigl{(}X|T\bigr{)}\,:=\,H_{\alpha}\bigl{(}X,T\bigr{)}-H_{\alpha} \bigl{(}T\bigr{)}\,=\,H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\odot K_{T}\bigr{)}-H_ {\alpha}(\frac{1}{n}K_{T})$ (6) $$

Specifically, it is shown that the defined conditional entropy $H_{\alpha}(X|T)$ is non-negative for every normalized kernel function $k_{X}$ and $k_{T}$. Furthermore, [] define the matrix-based mutual information $I_{\alpha}(X;T)$ as the difference between the defined conditional and marginal entropy which is shown to be non-negative given normalized kernel functions $k_{X}$ and $k_{T}$:

$$ $I_{\alpha}\bigl{(}X;T\bigr{)}\,:=\,H_{\alpha}\bigl{(}X\bigr{)}-H_{\alpha}\bigl {(}X|T\bigr{)}\,=\,H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\bigr{)}+H_{\alpha}\bigl{ (}\frac{1}{n}K_{T}\bigr{)}-H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\odot K_{T}\bigr{)}$ (7) $$

## 4 An Information-theoretic Diversity Quantification for Prompt-based Generative Models

We aim to extend the entropy-based diversity scores for unconditional generative models to conditional text-based generative models. Note that if we only evaluate the entropy score of the generated samples $x_{1},\ldots,x_{n}$, the evaluated score does not separate the diversity contributed by different prompts from the internal diversity of the model creating varying samples for similar prompts.

To separate out the effect of diverse input prompts on the variety of the generated samples $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}$, we propose applying the conditional entropy as formulated by [] for a general quantum information-theoretic setting and propose the following *order-$\alpha$ Conditional-Vendi Score*:

$$ $\displaystyle\mathrm{Conditional}\text{-}\mathrm{Vendi}_{\alpha}\bigl{(}x_{1}, \ldots,x_{n}\big{|}t_{1},\ldots,t_{n}\bigr{)}\,:=$ $\displaystyle\,\exp\Bigl{(}H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\odot K_{T}\bigr{ )}-H_{\alpha}\bigl{(}\frac{1}{n}K_{T}\bigr{)}\Bigr{)}$ $$

In the above, $K_{X}$ and $K_{T}$ denote the kernel matrices for the generated samples and input prompts, respectively. In addition to the proposed Conditional-Vendi score for measuring the internal diversity of the model, we propose the *Information-Vendi* score following the identity $I(X;T)=H(X)+H(T)-H(X,T)$ in classical information theory:

$$ $\displaystyle\mathrm{Information}\text{-}\mathrm{Vendi}_{\alpha}\bigl{(}x_{1}, \ldotp,x_{n};t_{1},\ldotp,t_{n}\bigr{)}:=$ $\displaystyle\exp\Bigl{(}H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\bigr{)}+H_{\alpha} \bigl{(}\frac{1}{n}K_{T}\bigr{)}-H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\odot K_{T} \bigr{)}\Bigr{)}$ $$

Note that the defined $\mathrm{Conditional}\text{-}\mathrm{Vendi}_{\alpha}$ and $\mathrm{Information}\text{-}\mathrm{Vendi}_{\alpha}$ lead to a decomposition of $\mathrm{Vendi}_{\alpha}$ score [] into the product of the following two terms:

$$ $\displaystyle\mathrm{Vendi}_{\alpha}\bigl{(}x_{1},\ldots,x_{n}\bigr{)}\>=\>$ $\displaystyle\mathrm{Conditional}\text{-}\mathrm{Vendi}_{\alpha}\bigl{(}x_{1}, \ldots,x_{n}\big{|}t_{1},\ldots,t_{n}\bigr{)}$ $\displaystyle\times\mathrm{Information}\text{-}\mathrm{Vendi}_{\alpha}\bigl{(} x_{1},\ldots,x_{n};t_{1},\ldots,t_{n}\bigr{)}.$ $$

In the above, $\mathrm{Vendi}_{\alpha}$ measures the diversity of generated data $x_{1},\ldots,x_{n}$, while the $\mathrm{Conditional}\text{-}\mathrm{Vendi}_{\alpha}$ score is a quantification of the internal diversity of the model that is not influenced by the variety of input prompts. Also, $\mathrm{Information}\text{-}\mathrm{Vendi}_{\alpha}$ can be viewed as a relevance score quantifying how the diversity of generated samples is statistically correlated with the diversity of text prompts. Therefore, the proposed decomposition leads to a mechanism for the internal diversity evaluation of conditional generative models.

Figure: Figure 2: Evaluated Conditional-Vendi and Vendi scores on dog samples in the ImageNet dataset. (Left Plot): we simulated prompts on "dog" pictures without specifying the dog breed, (Right Plot) we added the breed information for dogs to the prompts.
Refer to caption: x2.png

## 5 Statistical Interpretation of the Entropy-based Scores

In this section, we statistically interpret the defined Conditional-Vendi and Information-Vendi scores. First, we derive the statistic estimated from empirical samples by the entropy-based scores. According to the Schur product theorem, the Hadamard product $K_{X}\odot K_{T}$ of PSD kernel matrices $K_{X},\,K_{T}$ will also be a PSD kernel matrix. We note that the kernel matrix corresponds to the following feature map $\phi_{X,T}:\mathcal{X}\times\mathcal{T}\rightarrow\mathbb{R}^{d_{x}d_{t}}$ where $\otimes$ denotes the Kronecker product:

$$ $\phi_{X,T}\bigl{(}[x,t]\bigr{)}\,=\,\phi_{X}(x)\otimes\phi_{T}(t)$ $$

The above holds due to the identity $\bigl{\langle}\phi_{X,T}\bigl{(}[x,t]\bigr{)}\,,\,\phi_{X,T}\bigl{(}[x^{\prime
},t^{\prime}]\bigr{)}\bigr{\rangle}=k_{X}\bigl{(}x,x^{\prime}\bigr{)}k_{T}
\bigl{(}t,t^{\prime}\bigr{)}$. The following proposition formulates kernel-based conditional entropy and mutual information using $\phi_{X,T}$.

###### Proposition 1 .

Consider the kernel matrices $K_{X}$ for samples $x_{1},\ldots,x_{n}$ and $K_{T}$ for samples $t_{1},\ldots,t_{n}$. Then, $\frac{1}{n}K_{X}\odot K_{T}$ (used for defining joint entropy $H_{\alpha}(X,T)$) share the same non-zero eigenvalues with the following kernel covariance matrix:

$$ $\widehat{C}_{X,T}\,:=\,\frac{1}{n}\sum_{i=1}^{n}\phi_{X,T}\bigl{(}[x_{i},t_{i} ]\bigr{)}\phi_{X,T}\bigl{(}[x_{i},t_{i}]\bigr{)}^{\top}=\,\frac{1}{n}\sum_{i=1 }^{n}\Bigl{[}\phi_{X}\bigl{(}x_{i}\bigr{)}\phi_{X}\bigl{(}x_{i}\bigr{)}^{\top} \Bigr{]}\otimes\Bigl{[}\phi_{T}\bigl{(}t_{i}\bigr{)}\phi_{T}\bigl{(}t_{i}\bigr {)}^{\top}\Bigr{]}$ $$

###### Corollary 1 .

Consider the composite feature map $\phi_{X,T}$ and joint kernel covariance matrix $\widehat{C}_{X,T}$ defined above. Then, given marginal kernel covariance matrices $\widehat{C}_{X}=\frac{1}{n}\sum_{i=1}^{n}\phi_{X}(x_{i})\phi_{X}(x_{i})^{\top}
,\,\widehat{C}_{T}=\frac{1}{n}\sum_{i=1}^{n}\phi_{T}(t_{i})\phi_{T}(t_{i})^{\top}$, the following holds for the defined conditional entropy and mutual information:

$$ $\displaystyle H_{\alpha}(X|T)=H_{\alpha}(\widehat{C}_{X,T})-H_{\alpha}( \widehat{C}_{T}),\quad I_{\alpha}(X;T)=H_{\alpha}(\widehat{C}_{X})+H_{\alpha}( \widehat{C}_{T})-H_{\alpha}(\widehat{C}_{X,T})$ $$

###### Proof.

The proof is deferred to the Appendix. Note that $\mathrm{Conditional}\text{-}\mathrm{Vendi}_{\alpha}(x_{1},\ldotp,x_{n}|t_{1},
\ldotp,t_{n})=\exp\bigl{(}H_{\alpha}(X|T)\bigr{)}$ and $\mathrm{Information}\text{-}\mathrm{Vendi}_{\alpha}(x_{1},\ldotp,x_{n};t_{1},
\ldotp,t_{n})=\exp\bigl{(}I_{\alpha}(X;T)\bigr{)}$
∎

Corollary 1 shows that given the underlying covariance matrices $C_{X}=\mathbb{E}_{x\sim P_{X}}\bigl{[}\phi_{X}(x)\phi_{X}(x)^{\top}\bigr{]}$, $C_{T}=\mathbb{E}_{t\sim P_{T}}\bigl{[}\phi_{T}(t)\phi_{T}(t)^{\top}\bigr{]}$, and $C_{X,T}=\mathbb{E}_{(x,t)\sim P_{X,T}}\bigl{[}\phi_{X,T}([x,t])\phi_{X,T}([x,t
])^{\top}\bigr{]}$, the defined entropy-based scores converge to the following statistics when the sample size $n$ tends to infinity:

$$ $\widetilde{H}_{\alpha}(X|T)=H_{\alpha}({C}_{X,T})-H_{\alpha}({C}_{T}),\quad \widetilde{I}_{\alpha}(X;T)=H_{\alpha}({C}_{X})+H_{\alpha}({C}_{T})-H_{\alpha} ({C}_{X,T}).$ $$

Note that the entropy-based statistic $H_{\alpha}({C}_{X})$ represents the statistic estimated by the logarithm of the Vendi score defined in [].

Next, we prove that for a mixture text distribution $P_{T}$ where the text variable follows random mode $G\in\{1,\ldots,m\}$, the defined conditional entropy score aggregates the expectation of the unconditional entropy score $H(X|G=i)$over the $m$ text modes $1,\ldots,m$. This result shows that the conditional entropy can be interpreted as the aggregation of unconditional entropy for different types of text variable $T$, where each prompt follows the hidden group variable $G\in\{1,\ldots,m\}$. The hidden variable $G$ reflects the mode of the mixture distribution for text data, e.g. based on the subject or keywords of the text prompt.

###### Theorem 1 .

Consider the Gaussian kernel with bandwidth $\sigma$. Suppose $T$ follows a mixture distribution $\sum_{i=1}^{m}\omega_{i}P_{T,i}$ where $\omega_{i}$ denotes the weight of the $i$th component $P_{T,i}$ with mean vector $\mu_{i}$ and total variance $\mathbb{E}_{T\sim P_{T,i}}[\|T-\mu_{i}\|^{2}_{2}]=\sigma_{i}^{2}$. Given the aggregation map $f(z)=\exp((1-\alpha)z)$, for every order $\alpha\geq 2$, the matrix-based order-$\alpha$ conditional entropy satisfies the following inequality where $g(z)=\frac{\alpha}{\alpha-1}\log\bigl{(}\frac{1}{1-z/\|\bm{\omega}\|_{\alpha}}
\bigr{)}$ is an increasing scalar function with $g(0)=0$:

$$ $\displaystyle\biggl{|}\,\widetilde{H}_{\alpha}(X|T)\,-\,f^{-1}\Bigl{(}\sum_{i= 1}^{m}\frac{\omega_{i}^{\alpha}}{\sum_{j=1}^{m}\omega_{j}^{\alpha}}\,f\bigl{(} \widetilde{H}_{\alpha}(X|G=i)\bigr{)}\Bigr{)}\,\biggr{|}$ $\displaystyle\leq\;$ $\displaystyle 2\,g\Bigl{(}8\sum_{i=1}^{k}\omega_{i}\frac{\sigma^{2}_{i}}{ \sigma^{2}}+32\sum_{i=2}^{k}\sum_{j=1}^{i-1}\omega_{i}\exp\Bigl{(}\frac{-\|{ \mu}_{i}-{\mu}_{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}\Bigr{)}$ $$

The above theorem shows that if the text samples come from $m$ well-separated modes satisfying $\frac{\|{\mu}_{i}-{\mu}_{j}\|_{2}}{\sigma}\gg 1$ for every $i\neq j$ and $\frac{\mathbb{E}_{P_{T,i}}[\|T-\mu_{i}\|^{2}_{2}]}{\sigma^{2}}\ll 1$, then the defined conditional entropy score $H(X|T)$ aggregates the unconditional entropy score $H(X|G=i)$ given the prompt group. Here, the aggregation is the mean operation characterized by map $f_{\alpha}(z)=\exp((1-\alpha)z)$ for the order-$\alpha$ matrix-based Renyi entropy $H_{\alpha}$.

Based on Theorem 1, we propose a text-type-based diversity evaluation, where we restrict the evaluation of the prompt-based generative model to the prompts in the same group, i.e. the same mode in the mixture text distribution. To do this, we find the eigendirections corresponding to the text clusters by performing an eigendecomposition of text kernel matrix $\frac{1}{n}K_{T}=\sum_{i=1}^{n}\lambda_{i}v_{i}v_{i}^{\top}$ where $\lambda_{1}\geq\cdots\lambda_{n}$ are the sorted eigenvalues and $v_{1},\ldots,v_{n}$ are the sorted eigenvectors. Then, we note that the Hadamard product $\frac{1}{n}K_{X}\odot K_{T}$ used for joint entropy can be decomposed as:
$\frac{1}{n}K_{X}\odot K_{T}\,=\,\sum_{i=1}^{n}\lambda_{i}\bigl{(}K_{X}\odot v_
{i}v_{i}^{\top}\bigr{)}.$
As a result, to evaluate the diversity of the generation model, we apply eigendocomposition to each $K_{X}\odot v_{i}v_{i}^{\top}$, where $v_{i}$ marks the samples in group $i$, and find the sample indices with the maximum entries on the principal eigenvectors of $K_{X}\odot v_{i}v_{i}^{\top}$.

###### Proposition 1 .

###### Corollary 1 .

###### Proof.

###### Theorem 1 .

## 6 Numerical Results

We empirically evaluated the Conditional-Vendi and Information-Vendi scores for three types of conditional generative models: 1) text-to-image, 2) text-to-video generation, and 3) image-captioning models. For text-to-image models, we considered Flux [], Stable Diffusion 2.1 [], Stable Diffusion XL [], GigaGAN [], Kandinsky [], and PixArt [, ]. For text-to-video models, we tested VideoCrafter1 [], Show-1 [], and Open-Sora []. For image-captioning models, we evaluated BLIP [], GIT [], and GPT4o-mini [].

Embeddings used in the evaluation of generative models. Unlike standard embedding-based scores for text-to-image models such as CLIPScore [], which require the same embedding model for the text and generated image, the definitions of Conditional-Vendi and Information-Vendi allow different feature extractors for text and generated sample. In our experiments, we followed [, ], to use the DINOv2 [] embedding for image data. For text data, we used Gemini [] and CLIP [], and for video samples, following the video evaluation literature [, , ], we used I3D [].
To select the bandwidth parameter $\sigma$,
similar to [], we chose the Gaussian kernel bandwidth for each type of data as the smallest $\sigma$ that ensures a variance below 0.01 in the evaluated score over independent evaluations. We observed that for image data, $\sigma\in[20,30]$; for text data, $\sigma\in[0.1,0.8]$; and for video data, $\sigma\in[10,20]$ can satisfy this requirement.

Quantifying model-induced diversity via Conditional-Vendi. To illustrate how Conditional-Vendi correlates with the model-induced diversity, we considered a toy experiment with 10 different dog breeds from the ImageNet dataset [] as simulated outputs for a text-to-image model. We generated two sets of prompts using GPT4o []. In the first set, the breed of dog in the picture was not specified, while in the other one, the breed was explicitly mentioned. As shown in Figure 2, increasing the number of breeds sampled from the dataset led to the growth of the Vendi score, regardless of the text prompt. However, Conditional-Vendi only increased when the breed was not specified in the prompts, and in the second case where the breed had been included in the prompt, the score remained relatively constant, implying that the diversity in pictures mostly follows the text prompt. To repeat this observation for text-to-image models, we considered 10 types of animals generated by Stable Diffusion XL, as shown in Figure 3. Similar to the previous experiment, we found that Conditional-Vendi increased at a more rapid rate when the prompts did not specify the type of animal in the picture. In contrast, when the animal types were specified in the prompts, there was only a slight increase in the Conditional-Vendi value.

Figure: Figure 3: Evaluated Conditional-Vendi and Vendi scores on animal samples generated by Stable Diffusion-XL. (Left Plot): we do not specify the animal types in the prompt, (Right Plot) we specify the animal types in the prompt.
Refer to caption: x4.png

Figure: Figure 4: Comparing Conditional-Vendi and Information-Vendi of different text-to-image models.
Refer to caption: x6.png

Text-to-Image Model Evaluation:
In Figure 4, we compared Flux, Stable Diffusion XL, GigaGAN, and Kandinsky. As shown in the plot, we generated 30,000 samples using each model, and then clustered the prompts into $k$ groups, using k-means clustering on the Gemini embedding of text data, for different values of $k$. To simulate varying diversity across clusters, we assigned the image generated for the center of each cluster to all prompts within that cluster and measured the scores for different clusters. For example, when $k$=2000, we had 2,000 images for the $k$ clusters, with the text in each cluster paired with one of the images. As $k$ grows, we observed an increase in Information-Vendi, validating the fact that the images become more relevant to their prompts. Also, the Conditional-Vendi increased, as we expect the diversity of images to grow with more text clusters. Notably, while GigaGAN achieved a higher Vendi score, its Information-Vendi score was lower than that of SD-XL. This observation suggests that GigaGAN performs well at generating diverse outputs given prompts, but in terms of relevance, Flux and SD-XL are better. Our results align with conclusions made by [].

Measuring Conditional-Vendi across prompt types.
To measure Conditional-Vendi conditioned on the prompt type, we created 5000 prompts with different categories using GPT4o and generated the corresponding images with the text-to-image models. In Figure 5, top 2 groups of PixArt-$\alpha$ in terms of conditional entropy values are shown. We observed that "dog" text-based top 3 clusters of images looked more diverse than the image clusters for "airplane"-type prompts. Also, our evaluated Conditional-Vendi score of "dog" texts was significantly higher than that of the "airplane" class. We have tested other generative models in the appendix.

Figure: Figure 5: Quantifying image diversity for different clusters of text prompts. Images are generated using the PixArt-$\alpha$ model.
Refer to caption: x8.png

Text-to-Video Model Evaluation.
For the experiments on video data, to ensure the fairness of our evaluation, we used VBench samples [], which generated samples belong to the 8 content categories. In Figure 6, we used VideoCrafter-1, Show-1, and Open-Sora-1.2. We observed that VideoCrafter videos look less diverse and, in some cases, may not correlate significantly with the captions, when compared to Open-Sora. Confirming this observation, the Conditional-Vendi and Information-Vendi scores were lower for VideoCrafter than those for Open-Sora.

Figure: Figure 6: Measuring Conditional-Vendi and Information-Vendi for text-to-video models
Refer to caption: x9.png

Figure: Figure 7: Conditional-Vendi and Information-Vendi of image-captioning models for 3 image types
Refer to caption: x10.png

Image-Captioning Evaluation.
For image captioning, we used 10 classes from the ImageNet dataset as input for BLIP-2, GIT and GPT4o-mini. In Figure 7, we compared captions for the top three groups of images: gas pump, church, and cassette player. GIT generated more diverse captions compared to BLIP, which was confirmed by the Conditional-Vendi scores. On the other hand, GPT4o-mini generated longer and more detailed captions compared to GIT, which was also reflected in the evaluated Conditional-Vendi and Information-Vendi scores.

## 7 Conclusion

In this work, we proposed an evaluation score to measure the internal diversity of prompt-based generative models, isolating diversity that is not induced by variations in text prompts. The proposed method is based on a decomposition of unconditional matrix-based entropy scores, Vendi and RKE, into Conditional-Vendi and Information-Vendi components. From a theoretical perspective, we derived the kernel-based statistics estimated by these scores and demonstrated their connection to the expectation of unconditional entropy values given a fixed text prompt. In our experiments, we evaluated the proposed scores in multiple settings where the ground-truth ranking of model diversity and relevance was known, showing that the scores correlate well with the ground-truth rankings. A future direction is to apply the proposed scores to quantify biases in existing models regarding sample generation across different human ethnicities and genders. Additionally, using these scores as a regularization penalty to train more diverse prompt-based models is another interesting area for further exploration.

## Appendix A Proofs

### A.1 Proof of Proposition 1

First, we observe that for every $x,x^{\prime}\in\mathcal{X}$ and $t,t^{\prime}\in\mathcal{T}$, the following holds:

$$ $\displaystyle\phi_{X,T}([x,t])^{\top}\phi_{X,T}([x^{\prime},t^{\prime}])\,$ $\displaystyle=\,\bigl{(}\phi_{X}(x)\otimes\phi_{T}(t)\bigr{)}^{\top}\bigl{(} \phi_{X}(x^{\prime})\otimes\phi_{T}(t^{\prime})\bigr{)}$ $\displaystyle=\,\bigl{(}\phi_{X}(x)^{\top}\phi_{X}(x^{\prime})\bigr{)}\otimes \bigl{(}\phi_{T}(t)^{\top}\phi_{T}(t^{\prime})\bigr{)}$ $\displaystyle=\,k_{X}(x,x^{\prime})\otimes k_{T}(t,t^{\prime})$ $\displaystyle=\,k_{X}(x,x^{\prime})k_{T}(t,t^{\prime})$ $$

Therefore, the Hadamard product of kernel matrices $\frac{1}{n}K_{X}\odot K_{T}$ can be written as

$$ $\frac{1}{n}K_{X}\odot K_{T}=\frac{1}{n}\Phi_{X,T}\Phi_{X,T}^{\top}$ $$

in terms of the matrix of samples’ feature maps $\Phi_{X,T}\in\mathbb{R}^{n\times d_{x}d_{t}}$ with its $i$th row being $\phi_{X,T}([x_{i},t_{i}])$. We observe that the matrices $\frac{1}{n}\Phi_{X,T}\Phi_{X,T}^{\top}$ and $\frac{1}{n}\Phi_{X,T}^{\top}\Phi_{X,T}$ share the same non-zero eigenvalues, that are the square of the singular values of $\Phi_{X,T}$. Therefore, $\frac{1}{n}K_{X}\odot K_{T}$ has the same non-zero eigenvalues as the following matrix

$$ $\frac{1}{n}\Phi_{X,T}^{\top}\Phi_{X,T}=\frac{1}{n}\sum_{i=1}^{n}\phi_{X,T}([x_ {i},t_{i}])\phi_{X,T}([x_{i},t_{i}])^{\top}$ $$

which is the defined matrix $\widehat{C}_{X,T}$. Therefore, the proof of the proposition is complete.

### A.2 Proof of Corollary 1

As we showed in Proposition 1, the Hadarmard product $\frac{1}{n}K_{X}\odot K_{T}$ shares the same non-zero eigenvalues with $\widehat{C}_{X,T}$. Also, as noted by [], $\frac{1}{n}K_{X}$ and $\frac{1}{n}K_{T}$ have the same non-zero eigenvalues as of $\widehat{C}_{X}$ and $\widehat{C}_{T}$, respectively. Since the order-$\alpha$ matrix-based entropy is only a function of the input matrix’s non-zero eigenvalues (zero eigenvalues have no impact on the entropy value), we can conclude that

$$ $\displaystyle{H}_{\alpha}(X|T)$ $\displaystyle:=H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\odot K_{T}\bigr{)}-H_{\alpha }(\frac{1}{n}K_{T})$ $\displaystyle=H_{\alpha}(\widehat{C}_{X,T})-H_{\alpha}(\widehat{C}_{T}),$ $$

and also

$$ $\displaystyle{I}_{\alpha}(X;T)$ $\displaystyle:=H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\bigr{)}+H_{\alpha}\bigl{(} \frac{1}{n}K_{T}\bigr{)}-H_{\alpha}\bigl{(}\frac{1}{n}K_{X}\odot K_{T}\bigr{)}$ $\displaystyle=H_{\alpha}(\widehat{C}_{X})+H_{\alpha}(\widehat{C}_{T})-H_{ \alpha}(\widehat{C}_{X,T}).$ $$

### A.3 Proof of Theorem 1

To prove Theorem 1, we begin by showing the following lemma.

###### Lemma 1 .

Suppose that the kernel function $k$ and variable $T$ satisfy the assumptions in Theorem 1. Then, the following Frobenius norm bound holds for $C_{i}=\mathbb{E}\bigl{[}\phi_{X}(x)\phi_{X}(x)^{\top}|G=i\bigr{]}$ where $G\in\{1,\ldots,m\}$ is the cluster random variable for text $T$:

$$ $\displaystyle\Bigl{\|}C_{X\otimes T}-\sum_{i=1}^{m}\omega_{i}C_{i}\otimes\phi_ {T}(\mu_{i})\phi_{T}(\mu_{i})^{\top}\Bigr{\|}^{2}_{F}\leq\frac{\sum_{i=1}^{m}2 \omega_{i}\sigma_{i}^{2}}{\sigma^{2}}.$ $$

###### Proof.

To show this lemma, we define $T_{i}$ as a variable distributed as $P_{T|G=i}$. Then,

$$ $\displaystyle\Bigl{\|}C_{X\otimes T}-\sum_{i=1}^{m}\omega_{i}C_{i}\otimes\phi( \mu_{i})\phi(\mu_{i})^{\top}\Bigr{\|}^{2}_{F}$ $\displaystyle=\,$ $\displaystyle\Bigl{\|}\mathbb{E}\bigl{[}\phi_{X}(x)\phi_{X}(x)^{\top}\otimes \phi_{T}(t)\phi_{T}(t)^{\top}\bigr{]}-\sum_{i=1}^{m}\omega_{i}C_{i}\otimes\phi (\mu_{i})\phi(\mu_{i})^{\top}\Bigr{\|}^{2}_{F}$ $\displaystyle=\,$ $\displaystyle\Bigl{\|}\sum_{i=1}^{m}\omega_{i}\mathbb{E}\bigl{[}\phi_{X}(x) \phi_{X}(x)^{\top}\otimes\phi_{T}(t)\phi_{T}(t)^{\top}\big{|}G=i\bigr{]}-\sum_ {i=1}^{m}\omega_{i}C_{i}\otimes\phi(\mu_{i})\phi(\mu_{i})^{\top}\Bigr{\|}^{2}_ {F}$ $\displaystyle=\,$ $\displaystyle\Bigl{\|}\sum_{i=1}^{m}\omega_{i}\mathbb{E}\bigl{[}\phi_{X}(x) \phi_{X}(x)^{\top}\otimes\phi_{T}(t)\phi_{T}(t)^{\top}\big{|}G=i\bigr{]}-\sum_ {i=1}^{m}\omega_{i}\mathbb{E}\bigl{[}\phi_{X}(x)\phi_{X}(x)^{\top}\otimes\phi_ {T}(\mu_{i})\phi_{T}(\mu_{i})^{\top}\big{|}G=i\bigr{]}\Bigr{\|}^{2}_{F}$ $\displaystyle=\,$ $\displaystyle\Bigl{\|}\sum_{i=1}^{m}\omega_{i}\mathbb{E}\bigl{[}\phi_{X}(x) \phi_{X}(x)^{\top}\otimes\Bigl{(}\phi_{T}(t)\phi_{T}(t)^{\top}-\phi_{T}(\mu_{i })\phi_{T}(\mu_{i})^{\top}\Bigr{)}\big{|}G=i\bigr{]}\Bigr{\|}^{2}_{F}$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\mathbb{E}\Bigl{[}\Bigl{\|}\phi_{X}(x) \phi_{X}(x)^{\top}\otimes\Bigl{(}\phi_{T}(t)\phi_{T}(t)^{\top}-\phi_{T}(\mu_{i })\phi_{T}(\mu_{i})^{\top}\Bigr{)}\Bigr{\|}^{2}_{F}\big{|}G=i\Bigr{]}$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\mathbb{E}\Bigl{[}\Bigl{\|}\phi_{X}(x) \phi_{X}(x)^{\top}\Bigr{\|}^{2}_{F}\Bigl{\|}\phi_{T}(t)\phi_{T}(t)^{\top}-\phi _{T}(\mu_{i})\phi_{T}(\mu_{i})^{\top}\Bigr{\|}^{2}_{F}\big{|}G=i\Bigr{]}$ $\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\mathbb{E}\Bigl{[}\Bigl{\|}\phi_{T}(t) \phi_{T}(t)^{\top}-\phi_{T}(\mu_{i})\phi_{T}(\mu_{i})^{\top}\Bigr{\|}^{2}_{F} \big{|}G=i\Bigr{]}$ $\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\mathbb{E}\Bigl{[}2-2\exp\Bigl{(}\frac{- \|t-\mu_{i}\|_{2}^{2}}{\sigma^{2}}\Bigr{)}\big{|}G=i\Bigr{]}$ $\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{[}2-2\exp\Bigl{(}\frac{-\mathbb{E} \bigl{[}\|t-\mu_{i}\|_{2}^{2}\big{|}G=i\bigr{]}}{\sigma^{2}}\Bigr{)}\Bigr{]}$ $\displaystyle\stackrel{{\scriptstyle(f)}}{{\leq}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{[}2-2\exp\Bigl{(}\frac{-\sigma_{i}^ {2}}{\sigma^{2}}\Bigr{)}\Bigr{]}$ $\displaystyle\stackrel{{\scriptstyle(g)}}{{\leq}}\,$ $\displaystyle\sum_{i=1}^{m}2\omega_{i}\frac{\sigma_{i}^{2}}{\sigma^{2}}$ $$

In the above, (a) follows from Jensen’s inequality for the convex Frobenius-norm-squared function. (b) holds because $\|A\otimes B\|^{2}_{F}=\|A\|^{2}_{F}\|B\|^{2}_{F}$ for every matrices $A,\,B$. (c) comes from the normalized Gaussian kernel satisfying $\langle\phi_{T}(t),\phi_{T}(t)\rangle=k(t,t)=1$, resulting in $\|\phi_{T}(t)\phi_{T}(t)^{\top}\|^{2}_{F}=\mathrm{Tr}\bigl{(}\phi_{T}(t)\phi_{
T}(t)^{\top}\phi_{T}(t)\phi_{T}(t)^{\top}\bigr{)}=\mathrm{Tr}\bigl{(}\phi_{T}(
t)\phi_{T}(t)^{\top}\bigr{)}=1$. (d) follows from the Gaussian kernel definition, proving that $\phi_{T}(t)^{\top}\phi_{T}(\mu_{i})=\exp\bigl{(}-\|t-\mu_{i}\|^{2}_{2}/2\sigma
^{2}\bigr{)}$. (e) shows the application of Jensen’s inequality to the concave $s(z)=1-\exp(-z)$. (f) holds because $s(z)=1-\exp(-z)$ is a monotonically increasing function. Finally, (g) follows from the inequality $1-\exp(-z)\leq z$ for every scalar $z$. Therefore, the proof is complete.

∎

Next, we apply the Gram–Schmidt process to $\phi_{T}(\mu_{1}),\ldots,\phi_{T}(\mu_{m})$ to find orthogonal vectors $u_{1},\ldots,u_{m}$. We let $u_{1}=\phi_{T}(\mu_{1})$. Then, for every $2\leq i\leq m$, we define

$$ $\displaystyle u_{i}\,:=\,$ $\displaystyle\phi({\mu}_{i})-\sum_{j=1}^{i-1}\langle\phi(\mu_{i}),u_{j}\rangle u _{j}.$ $$

As a result, the following holds

$$ $\displaystyle\Bigl{\|}\sum_{i=1}^{m}\omega_{i}C_{i}\otimes\phi(\mu_{i})\phi( \mu_{i})^{\top}-\sum_{i=1}^{m}\omega_{i}C_{i}\otimes u_{i}u_{i}^{\top}\Bigr{\| }^{2}_{F}$ $\displaystyle=\,$ $\displaystyle\Bigl{\|}\sum_{i=1}^{m}\omega_{i}C_{i}\otimes\Bigl{(}\phi(\mu_{i} )\phi(\mu_{i})^{\top}-u_{i}u_{i}^{\top}\Bigr{)}\Bigr{\|}^{2}_{F}$ $\displaystyle\stackrel{{\scriptstyle(h)}}{{\leq}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{\|}C_{i}\otimes\Bigl{(}\phi(\mu_{i} )\phi(\mu_{i})^{\top}-u_{i}u_{i}^{\top}\Bigr{)}\Bigr{\|}^{2}_{F}$ $\displaystyle\stackrel{{\scriptstyle}}{{=}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{\|}C_{i}\Bigr{\|}^{2}_{F}\Bigl{\|} \phi(\mu_{i})\phi(\mu_{i})^{\top}-u_{i}u_{i}^{\top}\Bigr{\|}^{2}_{F}$ $\displaystyle\stackrel{{\scriptstyle(i)}}{{\leq}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{\|}\phi(\mu_{i})\phi(\mu_{i})^{\top }-u_{i}u_{i}^{\top}\Bigr{\|}^{2}_{F}$ $\displaystyle\stackrel{{\scriptstyle(j)}}{{=}}\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{(}1+\|u_{i}\|^{4}-2\bigl{(}u_{i}^{ \top}\phi_{T}(\mu_{i})\bigr{)}^{2}\Bigr{)}$ $\displaystyle\leq\,$ $\displaystyle\sum_{i=1}^{m}\omega_{i}\Bigl{(}2-2\bigl{(}u_{i}^{\top}\phi(\mu_{ i})\bigr{)}^{2}\Bigr{)}$ $\displaystyle=\,$ $\displaystyle 2\sum_{i=1}^{m}\omega_{i}\Bigl{(}1+u_{i}^{\top}\phi(\mu_{i}) \Bigr{)}\Bigl{(}1-u_{i}^{\top}\phi(\mu_{i})\Bigr{)}$ $\displaystyle\stackrel{{\scriptstyle(k)}}{{\leq}}\,$ $\displaystyle 4\sum_{i=1}^{m}\sum_{j=1}^{i-1}\omega_{i}\exp\Bigl{(}\frac{-\| \mu_{i}-\mu_{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}.$ $$

Here, (h) follows from the application of Jensen’s inequality for the convex Frobenius-norm-squared. (i) holds since the text kernel is normalized and $\langle\phi_{X}(x),\phi_{X}(x)\rangle=k_{X}(x,x)=1$, and therefore $\|C_{i}\|_{F}\leq\mathbb{E}[\|\phi_{X}(x)\|^{2}_{2}]=1$. (j) follows from the expansion $\|uu^{\top}-vv^{\top}\|^{2}_{F}=\|u\|^{4}_{2}+\|v\|^{4}_{2}-2\langle u,v
\rangle^{2}$. (k) holds because $u_{i}^{\top}\phi_{T}(\mu_{i})\leq 1$ and

$$ $\displaystyle u_{i}^{\top}\phi_{T}(\mu_{i})=1-\sum_{j=1}^{i-1}\langle\phi_{T}( \mu_{i}),u_{j}\rangle^{2}\geq 1-\sum_{j=1}^{i-1}\exp\bigl{(}\frac{-\|\mu_{i}- \mu_{j}\|^{2}_{2}}{\sigma^{2}}\bigr{)}.$ $$

Since we know that for every matrices $A,B\in\mathbb{R}^{d\times d}$, $\|A+B\|^{2}_{F}\leq 2\|A\|^{2}_{F}+2\|B\|^{2}_{F}$, the above results show that

$$ $\displaystyle\Bigl{\|}C_{X\otimes T}-\sum_{i=1}^{m}\omega_{i}C_{i}\otimes u_{i }u_{i}^{\top}\Bigr{\|}^{2}_{F}\leq\sum_{i=1}^{m}4\omega_{i}\frac{\sigma_{i}^{2 }}{\sigma^{2}}+\sum_{i=2}^{m}\sum_{j=1}^{i-1}8\omega_{i}\exp\Bigl{(}\frac{-\| \mu_{i}-\mu_{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}.$ $$

As a result, the Hoffman-Wielandt inequality shows that for the sorted eigenvalues vector $\bm{\lambda}$ of $C_{X\otimes T}$ and sorted eigenvalues vector $\widetilde{\bm{\lambda}}$ of $\sum_{i=1}^{m}\omega_{i}C_{i}\otimes u_{i}u_{i}^{\top}$ the following holds:

$$ $\displaystyle\bigl{\|}\bm{\lambda}-\widetilde{\bm{\lambda}}\bigr{\|}^{2}_{2}\,$ $\displaystyle\leq\,\Bigl{\|}C_{X\otimes T}-\sum_{i=1}^{m}\omega_{i}C_{i} \otimes u_{i}u_{i}^{\top}\Bigr{\|}^{2}_{F}$ $\displaystyle\leq\,\sum_{i=1}^{m}4\omega_{i}\frac{\sigma_{i}^{2}}{\sigma^{2}}+ \sum_{i=2}^{m}\sum_{j=1}^{i-1}8\omega_{i}\exp\Bigl{(}\frac{-\|\mu_{i}-\mu_{j} \|^{2}_{2}}{\sigma^{2}}\Bigr{)}.$ $$

Since $u_{1},\ldots,u_{m}$ are orthogonal vectors, the definition of Kronecker product implies that the eigenvalues of $\sum_{i=1}^{m}\omega_{i}C_{i}\otimes u_{i}u_{i}^{\top}$ will be the union of the eigenvalues of $\omega_{i}C_{i}\otimes u_{i}u_{i}^{\top}$ over $i\in\{1,\ldots,m\}$. On the other hand, we know that the non-zero eigenvalues of $\omega_{i}C_{i}\otimes u_{i}u_{i}^{\top}$ will be equal to the factor $\omega_{i}\|u_{i}\|^{2}_{2}$ times the eigenvalues of $C_{i}$. Also, we know that $1\geq\|u_{i}\|^{2}_{2}\geq 1-2\sum_{j=1}^{i-1}\exp(-\frac{\|\mu_{i}-\mu_{j}\|^
{2}_{2}}{\sigma^{2}})$. Consequently, we can show that for vector $\widehat{\lambda}_{x\otimes t}=\mathrm{Union}\bigl{(}\omega_{i}\,\mathrm{Eigs}
(C_{i}):\,i\in\{1,\ldots,m\}\bigr{)}$, we have the following for every $\alpha\geq 2$ and defined increasing function $g$ in Theorem 1

$$ $\displaystyle\Bigl{|}\widetilde{H}_{\alpha}(X,T)-\frac{1}{1-\alpha}\log\bigl{( }\|\widehat{\lambda}_{x\otimes t}\|^{\alpha}_{\alpha}\bigr{)}\Bigr{|}\,\leq\,$ $\displaystyle g\bigl{(}\|\widetilde{\lambda}_{x\otimes t}\|_{\alpha}-\| \widehat{\lambda}_{x\otimes t}\|_{\alpha}\bigr{)}$ $\displaystyle\leq\,$ $\displaystyle g\bigl{(}\|\mathrm{sort}\bigl{(}\widetilde{\lambda}_{x\otimes t} \bigr{)}-\mathrm{sort}\bigl{(}\widehat{\lambda}_{x\otimes t}\bigr{)}\|_{\alpha }\bigr{)}$ $\displaystyle\leq\,$ $\displaystyle g\bigl{(}\|\mathrm{sort}\bigl{(}\widetilde{\lambda}_{x\otimes t} \bigr{)}-\mathrm{sort}\bigl{(}\widehat{\lambda}_{x\otimes t}\bigr{)}\|_{2} \bigr{)}$ $\displaystyle\leq\,$ $\displaystyle g\Bigl{(}\sum_{i=1}^{m}4\omega_{i}\frac{\sigma_{i}^{2}}{\sigma^{ 2}}+\sum_{i=2}^{m}\sum_{j=1}^{i-1}16\omega_{i}\exp\Bigl{(}\frac{-\|\mu_{i}-\mu _{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}\Bigr{)}.$ $$

Note that the above proof holds for every marginal distribution on $X$, and we choose a deterministic constant $X=\mathbf{0}$, then the joint entropy reduces to the marginal entropy and the above inequality also shows the following:

$$ $\displaystyle\Bigl{|}\widetilde{H}_{\alpha}(T)-\frac{1}{1-\alpha}\log\bigl{(} \|[\omega_{1},\ldots,\omega_{m}]\|^{\alpha}_{\alpha}\bigr{)}\Bigr{|}\,\leq\,$ $\displaystyle g\Bigl{(}\sum_{i=1}^{m}4\omega_{i}\frac{\sigma_{i}^{2}}{\sigma^{ 2}}+\sum_{i=2}^{m}\sum_{j=1}^{i-1}16\omega_{i}\exp\Bigl{(}\frac{-\|\mu_{i}-\mu _{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}\Bigr{)}.$ $$

Therefore, following the Triangle inequality and the definition $\widetilde{H}_{\alpha}(X|T)=\widetilde{H}_{\alpha}(X,T)-\widetilde{H}_{\alpha}
(T)$, the previous two inequalities prove that

$$ $\displaystyle\Bigl{|}\widetilde{H}_{\alpha}(X|T)-\Bigl{(}\frac{1}{1-\alpha} \log\bigl{(}\|\widehat{\lambda}_{x\otimes t}\|^{\alpha}_{\alpha}\bigr{)}-\frac {1}{1-\alpha}\log\bigl{(}\|[\omega_{1},\ldots,\omega_{m}]\|^{\alpha}_{\alpha} \bigr{)}\Bigr{)}\Bigr{|}$ $\displaystyle\,\leq\,$ $\displaystyle 2g\Bigl{(}\sum_{i=1}^{m}4\omega_{i}\frac{\sigma_{i}^{2}}{\sigma^ {2}}+\sum_{i=2}^{m}\sum_{j=1}^{i-1}16\omega_{i}\exp\Bigl{(}\frac{-\|\mu_{i}- \mu_{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}\Bigr{)}.$ $$

On the other hand, we can simplify the above expression as

$$ $\displaystyle\frac{1}{1-\alpha}\log\bigl{(}\|\widehat{\lambda}_{x\otimes t}\|^ {\alpha}_{\alpha}\bigr{)}-\frac{1}{1-\alpha}\log\bigl{(}\|[\omega_{1},\ldots, \omega_{m}]\|^{\alpha}_{\alpha}\bigr{)}$ $\displaystyle=\,$ $\displaystyle\frac{1}{1-\alpha}\log\bigl{(}\sum_{i=1}^{m}\omega_{i}^{\alpha}\| {\lambda}_{C_{i}}\|^{\alpha}_{\alpha}\bigr{)}-\frac{1}{1-\alpha}\log\bigl{(} \sum_{i=1}^{m}\omega_{i}^{\alpha}\bigr{)}$ $\displaystyle=\,$ $\displaystyle\frac{1}{1-\alpha}\log\bigl{(}\sum_{i=1}^{m}\frac{\omega_{i}^{ \alpha}}{\sum_{j=1}^{m}\omega_{j}^{\alpha}}\|{\lambda}_{C_{i}}\|^{\alpha}_{ \alpha}\bigr{)}$ $$

Note that the definition $f_{\alpha}(t)=\exp((1-\alpha)t)$ implies that $f^{-1}_{\alpha}(z)=\frac{1}{1-\alpha}\log(z)$, which connects to the entropy definition as $H(X|G=i)=f^{-1}_{\alpha}(\|{\lambda}_{C_{i}}\|^{\alpha}_{\alpha})$. As a result, we can combine the previous two equations and complete the proof as:

$$ $\displaystyle\Bigl{|}\widetilde{H}_{\alpha}(X|T)-f^{-1}_{\alpha}\Bigl{(}\sum_{ i=1}^{m}\frac{\omega_{i}^{\alpha}}{\sum_{j=1}^{m}\omega_{j}^{\alpha}}f_{\alpha }\bigl{(}\widetilde{H}_{\alpha}(X|G=i)\bigr{)}\Bigr{)}\Bigr{|}$ $\displaystyle\,\leq\,$ $\displaystyle 2g\Bigl{(}\sum_{i=1}^{m}4\omega_{i}\frac{\sigma_{i}^{2}}{\sigma^ {2}}+\sum_{i=2}^{m}\sum_{j=1}^{i-1}16\omega_{i}\exp\Bigl{(}\frac{-\|\mu_{i}- \mu_{j}\|^{2}_{2}}{\sigma^{2}}\Bigr{)}\Bigr{)}.$ $$

###### Lemma 1 .

###### Proof.

## Appendix B Additional Numerical Results

### B.1 Correlation between prompts and generated output

To measure the correlation between text and image using Information-Vendi, we used MS-COCO captions to generate images with Stable Diffusion XL and Flux. We gradually substituted the generated images with random ones for the same prompts at different substitution rates. As the substitution rate increased, the correlation between the text and image pairs decreased. In Figure 8, we measured Information-Vendi at various substitution rates and observed that as the substitution rate increased, Information-Vendi decreased, demonstrating that our score can successfully measure the correlation between text and image.
Unlike other correlation metrics, such as CLIPScore, which require the same embedding for both text and image, our method places no such restriction. This allows for the use of different embeddings for text and image. Furthermore, our approach can be easily generalized to other conditional models, such as text-to-text or text-to-video generation.

Figure: Figure 8: Substituting images generated from models trained on MS-COCO dataset.
Refer to caption: x11.png

### B.2 Measuring Conditional-Vendi across prompt types

In this section, we conducted additional experiments similar to those in Figure 5. We created 5,000 prompts across different categories using GPT4o and generated corresponding images with text-to-image models.
We reported Conditional-Vendi for the top 3 groups in the text data on PixArt-$\alpha$, Stable Diffusion XL
text-to-image generative models.

As shown in Figure 9, Figure 10 and Figure 11, we observed the same behavior during these experiments: the Conditional-Vendi score for "dog" prompts was significantly higher than for the "airplane" and "sofa" categories. This observation suggests that the outputs of generative models are unbalanced when presented with different groups of text prompts.

Figure: Figure 9: Quantifying image diversity for different clusters of text prompts. Images are generated using the Stable Diffusion XL model.
Refer to caption: x13.png

Figure: Figure 10: Quantifying image diversity for different clusters of text prompts. Images are generated using the PixArt-$\alpha$ model.
Refer to caption: x14.png

Figure: Figure 11: Quantifying image diversity for different clusters of text prompts. Images are generated using the Flux model.
Refer to caption: x15.png

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 8
presentation: 4
questions: My opinion of the work is already high so I do not have many questions. However, in light of Figure 6, it seems that it would be possible to achieve a high conditional Vendi score by a model that has poor alignment with the prompt. Is there any way that the proposed conditional/information Vendi decomposition can handle this case?
soundness: 4
strengths: Strengths of this paper include:
- The proposed metrics occupy a clearly defined niche in the literature, providing a rich and informative way to evaluate prompt-based generative models. This is useful for better comparisons of generative models (even applicable across different modalities) and for better model development by elucidating the shortcomings of current models.
- Solid theoretical justification and empirical evidence are provided to support the proposed metrics.
- The paper is exceedingly well-written. Prose is clear and mathematical statements are presented in a precise yet intuitive way.
summary: This paper proposes a metric for evaluating the diversity of generative models. Unlike previous approaches which focus on unconditional generations, this paper uses an information-theoretic decomposition of the entropy into conditional entropy (measuring variation in generation when conditioned on a prompt) and mutual information (measuring the remaining variation not explained by the conditional entropy). The first term is expressed as a conditional Vendi score and the second as an information Vendi score. It is shown that the Vendi score is the product of these two, and so generative models with similar Vendi scores can be compared based on their conditional diversity given a prompt. Theoretical justfication for the proposed metrics is provided, and it is also shown how the metric can be estimated from finite samples. The proposed metrics are explored in different contexts, including text-to-image, text-to-video, and image-captioning models. The results correspond with the intuitive interpretations of the scores.
weaknesses: Weaknesses of this paper include:
- The proposed approach can be viewed as incremental, since it is the combination of two ideas: the Vendi score and matrix-based conditional entropy. However, the execution of the paper is very good and is targeted at a gap in the current literature, so I do not count this as a major weakness.
- The contextualization and interpretation of Theorem 1 could be improved.
- Qualitative results could be improved in some places. For example, showing some of the generations from the models in Figure 4 would help to better clarify the tradeoffs of the models in terms of conditional Vendi vs. information Vendi score.

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confidence: 5
contribution: 1
initial_rating: 3
presentation: 3
questions: No additional question other than the major weakness above.
soundness: 2
strengths: - The motivation for decomposing the diversity of model generation is clear, despite that a more fine-grained structural analysis would be preferred.
- The theoretical interpretation of the scores provides sufficient insight for the decomposition. 
- The paper is well-organized and logically fluent.
summary: The paper introduces an information-theoretic approach to evaluate the diversity of prompt-based generative models, focusing on decomposing the diversity into prompt-induced and model-induced components. The authors achieve this by decomposing the Vendi score into the Conditional-Vendi score, which measures internal diversity not related to prompt variations, and the Information-Vendi score, which assesses the statistical correlation between generated outputs and their corresponding prompts. The authors validate their methodology through various experiments on text-to-image, text-to-video, and image-captioning models.
weaknesses: While this paper tackles an important problem, the following weaknesses prevent it from being accepted.

- First of all, I think the paper lacks novelty in terms of the proposed method. The concept of conditional entropy $H_\alpha$ is nothing new, and the proposed conditional-Vendi score is simply its exponential version. If the paper is not contributing through new concepts or definitions, it should at least demonstrate some (new) important properties or use cases. Unfortunately, the only property discussed in the paper is its relevant stability when more diverse prompts are used, which, theoretically speaking, is expected. In such a case, the only novelty I can admit is that the paper demonstrates in experiments that the conditional entropy behaves normally in generative models. The authors are expected to provide a more comprehensive analysis of the given entropy and demonstrate a wider range of use cases if the concept itself is inherited from existing literature.
- Even if we consider only the stability property, the range of experiments in the paper is still limited. All experiments are simply measuring the scores with varying amounts of input clusters (groups), and the way that the authors make prompts diverse is also rigid. It is entirely unclear, based on existing experiments, that the proposed score can serve as an effective diversity measurement across a wide range of text prompts types. What's more, the authors do not show any support that the measure is accurate, except for this extremely intuitive "increasing curve". No golden metric, or even any proxy metric, has been introduced for comparison. This means that the "diversity measurement" claim is not convincing in the first place.
- Theoretically speaking, a large portion of the discussion about the selection of kernels is missing. From my understanding, since the authors only measure diversity from the generated data (rather than the underlying softmax distribution), the similarity metric used for text and images is very important and can largely change the metric. It is, therefore, necessary for the authors to carefully analyze the choice of kernels and make a more well-studied claim w.r.t. the proposed metric. However, notice that addressing this bullet is insufficient for making the paper reach the acceptance bar unless the first two weaknesses can be addressed.

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confidence: 4
contribution: 2
initial_rating: 3
presentation: 3
questions: (Contribution) As mentioned in Weakness, could the authors explain their contributions to prior works such as Giraldo et al (2014)?

 (Applications) For prompts that can be easily clustered, why don't we just average unconditional entropy for each cluster? For more general distribution, what's the practical meaning of the conditional entropy when the distribution $\mathcal{T}$ changes?
 (Numerical) Why can current numerical results illustrate the advantages of proposed metric?
soundness: 3
strengths: Originality: the problem of diversity quantification is indeed interesting and useful in generative models. 

Clarity: the paper is well-organized and straightforward to follow, with supportive numerical results and clearly stated details.

Overall the paper is a nice reading.
summary: This paper focuses on the diversity in prompt-based generative models. Specifically, the authors want to quantify the prompt-induced and model-induced diversity in samples separately. An information-theoretic approach for internal diversity quantification based on kernel-based conditional entropy of the generated data is discussed, where the Conditional-Vendi score and the Information-Vendi score are applied to measure the statistical relevance between the generated data and text prompts.
weaknesses: My major concern is that the contribution of this work is incremental. As stated by the paper in Section 3.2, the conditional entropy $H_\alpha(X|T)$ is already developed by Giraldo et al (2014), while the Conditional-Vendi score proposed by the authors is $\exp\left(H_\alpha(X|T)\right)$, which is essentially the same. In addition, existing literature has considered Renyi entropy of the probability model as the model's unconditional diversity score, and the idea of constructing conditional diversity score based on these results does not compose of sufficient contribution. Thus, I feel the contribution is incremental.

Another concern I have is the potential applications of the proposed diversity measure. In most scenarios mentioned by the paper, say statistical interpretation in Section 5 and some motivating examples, prompts can be easily clustered, so what about just calculate the unconditional entropy for each cluster of prompts, and then use some weighted average to combine them? I personally feel this is a naive but reasonable way to measure the conditional diversity of the model and should serve as a baseline. If prompts cannot be easily clustered and come from some complex distributions, then mathematically the conditional entropy would depend heavily on the distribution of the prompt distribution, the the concrete meaning of this measure is unclear to me. 

For numerical results, though they are clear to me, after reading it I can still not tell if conditional entropy is a good measure, as there is no baseline compared (which is reasonable as this is a relatively new problem, but the authors should at least consider some naive methods such as the one mentioned above, to see if there are some scenarios that their proposed method works while naive methods don't) and the meaning of values reported (say, figure 5-7) remains unclear to me. Why can these figures and values justify that the conditional entropy is a good measure of model's diversity?

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confidence: 3
contribution: 1
initial_rating: 3
presentation: 2
questions: 1. What's new of your conditional-Vendi compared with the conditional entropy proposed by Sanchez Giraldo et al., besides the exponential?
2. Follow which equation can the conditional-Vendi score be calculated? It's better to write an explicit algorithm, perhaps in the appendix.
3. What do you want to convey by Section 5 (the statistical interpretations)? Is it discussing the implementations, or just broader discussions? A better leading sentence may be necessary in the beginning.
soundness: 3
strengths: * The idea is sound to separate different sources of diversity, i.e., prompt-induced and model-induced.
* The proposed diversity metric is reasonable for model-induced diversity, and have the advantage that do not need repeated sampling from a single prompt.
summary: This paper focuses on distinguishing different sources for diversity evaluation of prompt-based generative models, i.e. prompt-induced diversity and model-induced diversity. Specifically, this paper proposed a new diversity evaluation metric called *condtional-Vendi* score, which is an extension of *Vendi*, but focuses on model-induced diversity rather than overall diversity. Combining with another proposed *information-Vendi* score, the *Vendi* score can be recovered by a multiplication. Experiments on image and text generation tasks show that condtional-Vendi is correlated with model-induced diversity.
weaknesses: The main weakness of this paper is its contribution. The proposed condtional-Vendi and information-Vendi score are just exponentials of the conditional entropy and mutual information defined by Sanchez Giraldo et al. (line 241, 246), and the calculation method seems to directly inherit from Vendi proposed by Friedman & Dieng (line 317). The combination is straightforward and I did not find any new challenges that is addressed by this paper. As a result, I view this paper as just an incremental work, and not enough for acceptance.

Meta-review: This work presents an information-theoretic approach to evaluate the diversity of prompt-based content generation. The proposed evaluation score defined in Section 4 consists of two parts: Conditional-Vendi and Information-Vendi. A statistical interpretation of the proposed score is presented in Section 5 (Theorem 1). The main criticism lies in the contribution. 3 out of the 4 reviewers think the proposed method is incremental at most and the proposed metric is derived from a simple transformation of existing ones. The reviewers also raise some questions on the numerical experiment design.

Decision: Reject