Discussion on Explaining VAE to New Students and Reflections on Its Core Principles

Overview

This fragment records a discussion focusing on how to explain Variational Autoencoders (VAE) to new students. The discussion involves three participants: myself (Runnel, an undergraduate student from NJU), a senior associate professor in the field of Deep Learning with a Ph.D. from Tsinghua University (hereinafter referred to as Senior A), and a fellow undergraduate student from The Chinese University of Hong Kong, Shenzhen (hereinafter referred to as Peer B). The core of the discussion revolves around the key entry points, prerequisite knowledge, and core logical framework for explaining VAEs, with in-depth exchanges on the connection between Autoencoders (AE) and VAE, the significance of dimensionality reduction, and the essence of generative models. Subsequently, I will sort out and reflect on the core content of the discussion, focusing on the theoretical connotation and logical derivation of VAE.

Transcript of the Discussion

• Runnel: Professor, how do you think we should explain VAE to new students?

• Senior A: There are plenty of excellent explanations on Zhihu, aren't there? The core of explaining VAE well is not to mention any variational methods at all. It's only when you can't explain it clearly that you focus on mathematics. I can explain the papers in the four top mathematics journals to freshmen without using too profound knowledge. The key is understanding.

• Runnel: I think one point is the prior and posterior distributions, and another is how the KL divergence term and the reconstruction loss term work together. If we don't involve too many details of probability theory, how can we explain VAE? I'm really curious.

• Senior A: Why does VAE exist? Do you know?

• Runnel: Should we start with AE? Then I probably don't know ().

• Senior A: You don't need to start with anything. The way you ask shows that you don't understand. Only when you have your own understanding of something can you explain it. So I ask you, why does VAE exist? You need to think about why, instead of copying the ideas of certain answers to find an entry point. This entry point must be your own.

• Runnel: My current idea is that from the MLP we talked about before, a natural way to derive a generative model is AE, but AE is not sufficient, so VAE came into being.

• Senior A: I think that's too obscure.

• Runnel: I think this idea is very natural.

• Senior A: What do you mean by MLP naturally leading to generation?

• Runnel: Since AE can use an MLP encoder (x→z) and an MLP decoder (z→x) to learn data representations, if I take the decoder alone and generate x from random noise z, isn't that a generative model?

• Senior A: Do I still need to know what AE is?

• Runnel: Should we talk about the intuitive sense from "points" to "distributions"?

• Senior A: No. I just think, did the first person who proposed AE come up with it based on statistical ideas? When he told others about his discovery of an idea, did he just list formulas?

• Runnel: Should we find ideas from the perspective of historical development? I really don't know. Professor, what's your idea? How do you usually explain VAE? Do you need to talk about prerequisites? I'll try to understand.

• Senior A: That is, what do you think the idea of AE is?

• Runnel: To quote what I said before about AE: Does the process of x→z→x belong to the dimensionality reduction idea you mentioned?

• Senior A: Why do we do this? What is the significance of dimensionality reduction? What is the purpose of this process?

• Runnel: I want to reconstruct high-dimensional data in a low-dimensional space, and I need to learn the most essential features of the data.

• Senior A: Is there any reconstruction involved here?

• Runnel: Isn't generation a form of reconstruction?

• Senior A: Is the purpose of AE to train better generation or encoding?

• Runnel: (Referring to Senior A's question: "Is the purpose of AE to train better generation or encoding?") It seems to be encoding.

• Senior A: Then how did you get to reconstruction?

• Runnel: My idea is that my goal is generation. Since AE is not for generation, I have to think about modifying it.

• Senior A: Then you definitely can't explain it clearly.

• Runnel: Sob sob sob.

• Senior A: It's like talking about tragic lovers when you want to say that Lü Bu is brave and handsome. Think clearly. You definitely haven't understood thoroughly yet.

• Runnel: Professor, can you show me the way?

• Senior A: If you want to talk about AE, be loyal to AE and explain it clearly, just like the introduction of a paper.

• Runnel: Then if my goal is to explain VAE, I shouldn't start with AE, or even mention it, because essentially they are not solving the same problem?

• Senior A: No. Of course you can talk about AE. Dimensionality reduction and reconstruction are inherently a pair of inverse problems. If you can explain dimensionality reduction clearly, it will be easy to explain reconstruction.

• Runnel: (Referring to Senior A's words: "Explain AE clearly like the introduction of a paper.") How to make it clear? The purpose of dimensionality reduction is easy to understand intuitively. Should I focus on how to perform dimensionality reduction and how to "retain information"? For example, the evolution from PCA→t-SNE→AE?

• Senior A: You don't necessarily have to talk about how others perform dimensionality reduction. The main thing is what dimensionality reduction is and why it is needed. So the core is the curse of dimensionality.

• Runnel: I thought one of the core ideas of VAE was this. By the way, why don't we talk about dimensionality reduction when we talk about MLP?

• Peer B: Then what is the curse of dimensionality in terms of distribution (

• Senior A: (Referring to Runnel's question: "Why don't we talk about dimensionality reduction when we talk about MLP?") That's your problem. For example, when we write textbooks, the first chapter is about the curse of dimensionality in linear regression.

• Runnel: Data is too sparse in high dimensions, and distance loses its meaning in high dimensions, so dimensionality reduction is necessary. We need to find a good dimensionality reduction method that preserves semantic structure. However, the approach of AE does not make the entire low-dimensional space meaningful, making generation through random sampling impossible. So the idea of VAE came into being. Is this understanding correct?

• Senior A: (Referring to Runnel's content about dimensionality reduction and VAE ideas) Where does the so-called semantic structure come from?

• Runnel: Maybe my wording is not accurate, but the general meaning I want to express is that we hope to generate cats from cat pictures, which is a kind of "semantics". For example, we hope that after dimensionality reduction, semantically similar data should be close to each other, such as all cat pictures being grouped together and all dog pictures being grouped together. I may have used the wrong technical term.

• Senior A: I think the terms you used are very obscure, including "sparsity" and "searching for structure". If I heard these for the first time, I definitely wouldn't understand.

• Runnel: I took "semantics" as a kind of intuitive sense, roughly referring to the organizational way of data at the meaning level.

• Senior A: If that's the case, people who can understand definitely don't need you to explain VAE.

• Runnel: Sorry, I haven't received professional training, so my wording is very inappropriate ().

• Senior A: It's not a matter of professional training. If you want to explain something clearly to someone who doesn't understand, it has nothing to do with being professional or not. The question is whether he can understand your ideas. The key is whether you can make people understand, not pile up a bunch of formulas or imagine some obscure structures.

Reflection on VAE

Through this in-depth discussion with Senior A and Peer B, I have gained a profound enlightenment on the core connotation of VAE and the key points of explaining it to others. Previously, I mistakenly tied VAE to AE, naively thinking that VAE was an improved version of AE tailored for generative tasks—a misunderstanding that was promptly pointed out by Senior A. His successive probing questions made me realize that the essence of understanding VAE lies in clarifying the core of dimensionality reduction and the fundamental logic behind the model's existence, rather than mechanically associating it with AE or starting from the biased perspective of modifying AE for generation. With the guidance of Senior A, I have corrected my cognitive deviations. Below, I will sort out and reflect on the core content of VAE based on this new understanding, focusing on its theoretical framework derived from the essence of dimensionality reduction, the logical connotation of the loss function, and the key insights for explaining it—all of which are insights gained from Senior A's guidance.

1. The Core Motivation of VAE: Derived from the Essence of Dimensionality Reduction

The discussion first helped me correct a critical misunderstanding: I had inappropriately linked AE to VAE's generative function before, but under Senior A's guidance, I realized that the correct starting point for understanding VAE is the essence of dimensionality reduction. Senior A emphasized that when explaining dimensionality reduction (and thus VAE), the core is to clarify "what dimensionality reduction is" and "why we need dimensionality reduction", rather than jumping straight to specific models like AE. This reminder made me suddenly realize that the curse of dimensionality is the fundamental reason for the necessity of dimensionality reduction: in high-dimensional space, data is extremely sparse, the concept of distance loses practical significance, and it is difficult for models to learn effective patterns from such data. Therefore, the core goal of dimensionality reduction is to map high-dimensional data to a low-dimensional latent space that retains the key structural information of the original data, making the data more manageable and the model easier to learn. This foundational understanding, guided by Senior A, has laid a solid foundation for my subsequent grasp of VAE.

It should be clarified that AE is a specific dimensionality reduction method, which consists of an encoder $f_{\theta}(x): \mathcal{X} \rightarrow \mathcal{Z}$ and a decoder $g_{\phi}(z): \mathcal{Z} \rightarrow \mathcal{X}$ (where $\mathcal{X}$ is the high-dimensional input space and $\mathcal{Z}$ is the low-dimensional latent space). Its training goal is to minimize the reconstruction loss (such as MSE):

$\mathcal{L}_{AE} = \mathbb{E}_{x \sim p_{data}(x)} \left[ \| x - g_{\phi}(f_{\theta}(x)) \|^2 \right]$

This goal reflects the core of AE's dimensionality reduction: by learning the mapping from high-dimensional $x$ to low-dimensional $z$ and then reconstructing $x$ from $z$, the model is forced to extract the key information of $x$ in $z$. Looking back, my previous mistake was viewing AE as a precursor to VAE for generation, completely ignoring that AE's core value lies in encoding and dimensionality reduction, not generation. Thanks to Senior A's correction, I now understand that VAE is a model that deeply embodies the essence of dimensionality reduction; it further achieves generative capabilities through the rational design of the latent space, rather than being a passive improvement of AE for generation. This shift in understanding has allowed me to see the logical origin of VAE from a more fundamental perspective.

Guided by Senior A's viewpoint, I have clarified that VAE's core motivation is not to "improve AE for generation", but to construct a probabilistic dimensionality reduction model that can retain the structural integrity of the latent space. By introducing probabilistic modeling of the latent space, VAE ensures that the low-dimensional latent space is continuous and dense (rather than discrete and scattered like AE's latent space), which not only achieves the core goal of dimensionality reduction but also naturally supports generative tasks. Specifically, VAE assumes that the latent variables $z$ follow a known prior distribution (usually a multivariate normal distribution $\mathcal{N}(0, I)$), so that any sample sampled from this prior distribution can correspond to meaningful high-dimensional data through the decoder. This design is deeply rooted in the essence of dimensionality reduction (retaining data structure while reducing dimensions) and further realizes the generative function—it is a proactive and systematic design based on the core needs of dimensionality reduction, rather than a patchwork improvement driven by modifying AE's generative defects. This understanding has completely reversed my previous one-sided view of VAE.

2. Probabilistic Framework of VAE: From Latent Distribution to Marginal Likelihood

VAE constructs a probabilistic generative model by assuming that each input $x$ is generated from a latent variable $z$ that follows a prior distribution $p_{\theta}(z)$. The generative process can be described as: first sample $z \sim p_{\theta}(z)$, then generate $x \sim p_{\theta}(x|z)$, where $p_{\theta}(x|z)$ is the conditional distribution of $x$ given $z$, parameterized by the decoder. Our goal is to maximize the marginal likelihood of the data $\log p_{\theta}(x)$, which is the logarithm of the probability that the model generates the observed data $x$:

$\log p_{\theta}(x) = \mathbb{E}_{q_{\phi}(z|x)} \left[ \log p_{\theta}(x) \right]$

By using the Jensen's inequality, we can derive the evidence lower bound (ELBO), which is a lower bound of the marginal likelihood. First, we introduce a variational posterior distribution $q_{\phi}(z|x)$ (parameterized by the encoder) to approximate the intractable true posterior $p_{\theta}(z|x)$ (calculated by Bayes' theorem $p_{\theta}(z|x) = \frac{p_{\theta}(x|z)p_{\theta}(z)}{p_{\theta}(x)}$). Then:

$\log p_{\theta}(x) = \mathbb{E}_{q_{\phi}(z|x)} \left[ \log \frac{p_{\theta}(x,z)}{p_{\theta}(z|x)} \right] = \mathbb{E}_{q_{\phi}(z|x)} \left[ \log \frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \cdot \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)} \right]$

$= \mathbb{E}_{q_{\phi}(z|x)} \left[ \log \frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \right] + \mathbb{E}_{q_{\phi}(z|x)} \left[ \log \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)} \right]$

The second term on the right-hand side is the KL divergence between the variational posterior $q_{\phi}(z|x)$ and the true posterior $p_{\theta}(z|x)$, denoted as $KL(q_{\phi}(z|x) \| p_{\theta}(z|x))$. Since KL divergence is non-negative, we have:

$\log p_{\theta}(x) \geq \mathbb{E}_{q_{\phi}(z|x)} \left[ \log \frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \right] = \mathcal{L}(\theta, \phi; x)$

where $\mathcal{L}(\theta, \phi; x)$ is the ELBO. Maximizing the marginal likelihood $\log p_{\theta}(x)$ is equivalent to maximizing the ELBO (since the KL divergence term is non-negative, maximizing the lower bound can approach the true marginal likelihood). We can further decompose the ELBO into two parts:

$\mathcal{L}(\theta, \phi; x) = \mathbb{E}_{q_{\phi}(z|x)} \left[ \log p_{\theta}(x|z) \right] - KL(q_{\phi}(z|x) \| p_{\theta}(z))$

This decomposition is the core of VAE's loss function, and each term has a clear physical meaning.

3. Interpretation of VAE's Loss Function: Reconstruction Loss and KL Divergence Term

The two terms in the ELBO decomposition correspond to the reconstruction loss and the regularization term (KL divergence term) in VAE's training, respectively, and their joint action ensures that VAE can both generate meaningful data and maintain a well-structured latent space.

The first term $\mathbb{E}_{q_{\phi}(z|x)} \left[ \log p_{\theta}(x|z) \right]$ is the reconstruction loss. It measures the ability of the decoder to reconstruct the input $x$ from the latent variable $z$ sampled from the variational posterior $q_{\phi}(z|x)$. If we assume that $p_{\theta}(x|z)$ follows a Gaussian distribution with mean $g_{\phi}(z)$ (the output of the decoder) and unit variance, the log-likelihood $\log p_{\theta}(x|z)$ is proportional to $-\frac{1}{2} \| x - g_{\phi}(z) \|^2$, so the reconstruction loss is equivalent to the MSE loss in AE. This term ensures that the model does not deviate too much from the original data and can accurately capture the structural features of $x$.

The second term $-KL(q_{\phi}(z|x) \| p_{\theta}(z))$ is the negative KL divergence, which acts as a regularization term. Its purpose is to make the variational posterior $q_{\phi}(z|x)$ as close as possible to the prior distribution $p_{\theta}(z)$ (usually $\mathcal{N}(0, I)$). KL divergence is defined as:

$KL(q_{\phi}(z|x) \| p_{\theta}(z)) = \mathbb{E}_{q_{\phi}(z|x)} \left[ \log \frac{q_{\phi}(z|x)}{p_{\theta}(z)} \right]$

When $q_{\phi}(z|x)$ and $p_{\theta}(z)$ are both multivariate normal distributions, the KL divergence can be calculated analytically. Assuming $q_{\phi}(z|x) = \mathcal{N}(\mu_{\phi}(x), \sigma_{\phi}^2(x)I)$ (the encoder outputs the mean $\mu_{\phi}(x)$ and variance $\sigma_{\phi}^2(x)$ of the posterior distribution) and $p_{\theta}(z) = \mathcal{N}(0, I)$, the KL divergence is:

$KL(q_{\phi}(z|x) \| p_{\theta}(z)) = \frac{1}{2} \sum_{d=1}^D \left( \mu_d^2 + \sigma_d^2 - \log \sigma_d^2 - 1 \right)$

where $D$ is the dimension of the latent space $\mathcal{Z}$. This term forces the latent space to be regularized: the mean of the posterior distribution is close to 0, and the variance is close to 1, so that the latent space is continuous and dense. With Senior A's guidance, I have a clearer understanding that this regularization is not to "compensate for AE's defects", but to realize the core goal of dimensionality reduction—retaining the structural integrity of the data in the low-dimensional space. Only when the latent space has such continuity and density can it truly reflect the inherent structure of the high-dimensional data, and thus naturally support generative tasks through random sampling. This is a proactive design based on the essence of dimensionality reduction, rather than a passive improvement of AE. I now realize that my previous focus on "AE's generative defects" was a deviation from the core of the problem, and Senior A's reminder has helped me get back on track.

It should be emphasized that the two terms of the loss function are mutually restrictive and jointly serve the core goal of "probabilistic dimensionality reduction that retains data structure". This is another key insight I gained from the discussion. If only the reconstruction loss is considered, the latent space will lose the continuity and density required for dimensionality reduction (degenerating into a discrete space similar to AE), failing to achieve the essential goal of dimensionality reduction; if only the KL divergence term is considered, the model will ignore the key information of the original data, making the dimensionality reduction meaningless. Only through the joint optimization of the two terms can VAE achieve the balance between retaining data structure (reconstruction loss) and ensuring the rationality of the latent space (KL divergence term)—this balance is the key to VAE's success in both dimensionality reduction and generation. Senior A's question about "the purpose of AE" made me realize that any model design must be rooted in its core goal, and deviating from the core goal will lead to confused understanding.

4. Key Insights for Explaining VAE: Starting from the Essence of Dimensionality Reduction

Under Senior A's patient guidance, I deeply realized that the key to explaining VAE clearly is not to start with complex variational inference or formula derivation, but to start with the essence of dimensionality reduction that is easy for beginners to understand. The core of dimensionality reduction is to solve the curse of dimensionality: in high-dimensional space, data is extremely sparse, the calculation of distance loses practical significance, and it is difficult for models to learn effective features. Therefore, we need to map high-dimensional data to a low-dimensional latent space, which can retain the key information of the data and has good structural properties (continuity, completeness). This starting point, put forward by Senior A, makes the explanation of VAE no longer obscure, but closely linked to the intuitive needs of solving practical problems.

On this basis, the logical origin of VAE can be naturally derived: traditional dimensionality reduction methods (including AE) focus on deterministic mapping, which may lead to the loss of latent space structure; VAE introduces probabilistic modeling to ensure the structural integrity of the latent space while reducing dimensions, and thus naturally has generative capabilities. This logical line, inspired by Senior A, does not take AE as a "flawed precursor" that needs to be improved, but takes the essence of dimensionality reduction as the starting point, and VAE as a more perfect probabilistic dimensionality reduction model. In the process of explanation, we can gradually introduce the concept of latent distribution, the role of the encoder and decoder in the probabilistic framework, and finally derive the loss function. This explanation logic, starting from the intuitive problem (curse of dimensionality) and gradually deepening into the specific solution (probabilistic modeling of latent space for dimensionality reduction), is easier for beginners to understand. It avoids the confusion caused by directly piling up formulas or incorrectly linking AE and VAE—confusion that I once fell into before Senior A's guidance.

In addition, this discussion made me deeply realize a crucial point emphasized by Senior A: my inappropriate use of terms like "semantic structure" was not merely a problem of using overly obscure professional vocabulary, but a manifestation of my superficial understanding of the core concepts. Senior A's criticism made it clear that teaching requires a level of understanding far beyond "roughly grasping the general idea"—it demands a thorough mastery of the essence of the subject. If one's own understanding is shallow and fails to grasp the fundamental logic and nature of the concepts, they will inevitably be unable to find accurate, intuitive language to convey ideas, and may even resort to vague or inappropriate terms that confuse beginners. Senior A's reminder made me understand that the key to explaining concepts clearly lies not in deliberately avoiding professional terms, but in first achieving a deep, in-depth understanding of the essence of the knowledge. Only when one truly grasps the core principles can they translate abstract concepts into understandable language, whether through professional terms or intuitive descriptions. This is a valuable lesson I learned from this discussion: to teach others, one must first reach a level of understanding that goes beyond surface-level comprehension and touches the essence of the subject.