<!-- Use this to grab and git add all images in a markdown file: grep img lecture21.md | cut -d\" -f6 | xargs git add --> <style> .container { display: flex; justify-content: center; align-items: center; } .col {flex: 1;} </style> # CS 530 - Lecture 21 ## Features and Latent Spaces Bernhard Firner 2026-04-14 --- ## What is AI? * Let's get philosophical for a moment * What is it that makes artificial intelligence a good solution to our problems? * And how can we make better use of our existing techniques? * We have a few lectures left, which should be just enough time to dig into this * Today we'll survey a few ideas, and then we'll dig into them more --- ## What is Knowledge? * Applying the parameters of a trained neural network means transforming the input into a desired output * Outputs are varied * Sometimes we get a single value * Other times we get class probabilities * And we could even transform one image into another image * The weights and biases of the neural network have knowledge, but what does that mean? --- ## Knowledge as Compression * [Shannon's Source Coding Theorem](https://en.wikipedia.org/wiki/Shannon's_source_coding_theorem) tells us that, when we compress i.i.d. data, the minimal compression of information scales with the entropy * Conversely, if what we are compressing is not i.i.d, [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity) is a better estimate * The Wikipedia summary is that the Kolmogorov complexity is the length of a shortest computer program that produces the object * We can make a decent definition by saying knowledge ignores noise and compresses non-noise * Learning noise is, after all, futile * It would be equivalent to learn its statistics and simply make new noise later --- ## 3 Examples * Let's look at three examples that neural network demonstrate compression of non-i.i.d. information * From simplest to most complicated (and perhaps least convincing to most convincing) * Feature Map Segmentation * Generative Models * Puzzle solving --- ## Feature Maps * If you train a classifier and cut off its head, you will find a feature map * This could be from a convnet, a transformer, or some other architecture * Those features are amenable to clustering via KNN, and can work to classify new, unseen image types * As in [Fast Incremental Learning for Off-Road Robot Navigation](https://arxiv.org/abs/1606.08057) * What information goes into a feature maps? And what information is dropped? --- ## Single Features * We can look at single feature maps (or the features from a self-attention head) * These are recognizable as single object types * Notice how the rest of the image has most features removed? * The "signal" is preserved, the "noise" is removed * Noise removal is a basic part of compression <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/dino_attention_maps.png" /> <br/> <small>See DINO results <a href="https://github.com/facebookresearch/dino">github page</a>.</small> </div> --- ## Training Method * Maybe you aren't impressed, but those features came from a model trained without any labels * [Emerging Properties in Self-Supervised Vision Transformers](https://openaccess.thecvf.com/content/ICCV2021/html/Caron_Emerging_Properties_in_Self-Supervised_Vision_Transformers_ICCV_2021_paper) (from 2021) * This paper introduced a self-supervision learning technique named DINO * Also pointed out that: * 1: unsupervised features contain scene layout and object boundaries * 2: unsupervised features can simply be k nearest neighbor clustered to make a classifier --- ## Self-Distillation * Why would the model learn such good semantic features without any labels? * This learning process asked the model to produce a K dimensional feature vector * The loss was how well the features matched another vector made from a different view of the image * The second features came from a teacher network * The teacher itself is a slow exponential moving average of the student --- ## Knowledge Distillation * The feature vector comparison is a scaled dot product similarity of the projected vectors * $similarity(u, v) = \frac{u^Tv}{||u||~||v||}$ * $s_{u,v}$ for short * The full loss is a softmax run over all similarities within the batch * $\ell(i, j) = -log\frac{exp(s_{i,j}/\tau)}{\sum\limits_{k=1}^{2N}\mathbb{1}_{[k\neq i]}exp({s_i,k} / \tau)}$ * $\tau$ is used as a temperature (hyperparameter) to control the rate of change --- ## Analogy and Result * Think of that loss as being asked to pick someone out of a lineup * You may have seen the person before, but they will not look the same * In order to make a match, you need to know something fundamental about human faces and bodies * The NN version is a bit harder; you may be asked to match to people given an image of a foot and a hand * The only way to do that well is to "project" features into a space where all of that noise is ignored and only the signal is present --- ## Feature Quality * The features have no labels * The best way to minimize loss is for the model to distinguish between object types * The authors find this lead to clean features with no spurious correlations * What is a suprious correlation? * There are waves, so the image has a boat * There is sky, so the image is a bird --- ## Example Features <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/EmergingProperties_figure4.png" /> <br/> <small>See <a href="https://openaccess.thecvf.com/content/ICCV2021/papers/Caron_Emerging_Properties_in_Self-Supervised_Vision_Transformers_ICCV_2021_paper.pdf">the DINO paper</a>, figure 4.</small> </div> --- ## Direct Applications * We could push that farther, using the raw feature maps for different applications * Semantic segmentation * Localization (and thus tracking) * Tracking * Once we know that an object is present, we can build a tracker for it * Since the self-supervised features should be robust to occlusion and rotation, this can be robust --- ## Segmentation * Eigenvalues of the feature maps plus the original image immediately yield segmentation * See [Deep Spectral Methods: A Surprisingly Strong Baseline for Unsupervised Semantic Segmentation and Localization](https://openaccess.thecvf.com/content/CVPR2022/html/Melas-Kyriazi_Deep_Spectral_Methods_A_Surprisingly_Strong_Baseline_for_Unsupervised_Semantic_CVPR_2022_paper.html) from CVPR in 2022 * Also the [github page](https://github.com/lukemelas/deep-spectral-segmentation) * It is fairly intuitive that feature masks can be used for segmentation * Eigenvalues are also used in PCA * Which can also be used as a type of compression, discarding components will less information --- ## Generative Models * So feature maps capture structure and give us an easy way to discard noise * But is what is left behind a compressed version of the original dataset? * Let's briefly discuss a type of generative model * Generative Adversarial Network, or GAN --- ## GANs * Usually, GANs consist of two networks * A *generator*, which creates synthetic data samples * A *discriminator*, which judges if an input is synthetic or real * That generator should sound reminiscent to [Kolmogorov complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity) * Kolmogorov complexity is the length of a shortest computer program that produces the object --- ## Loss Equation * [The Relativistic Discriminator: a Key Element Missing from Standard GAN](https://arxiv.org/pdf/1807.00734) introduces an improved GAN loss equation * Let's say $D(x) = sigmoid(C(x))$ * D is the discriminator, which classifies x as real or fake * C is a critic, just outputting a continuous value of the inputs "realness" --- ## Learning * The relativistic loss is $D(\tilde{x}) = sigmoid(C(x_{real} - C(x_{fake}))$ * The discriminator attempts to maximize this * The generator attempts to maximize the opposite * $D(\tilde{x}) = sigmoid(C(x_{fake} - C(x_{real}))$ * Thus the Discriminator and Generator compete to fool C --- ## Generator as Compressor * Imagine if the generator could create any possible image of a certain class * It would be an incredibly effective compressor of that class, right? * Generators from GANs are not quite that powerful, but they are impressive * And what do they need as input? Usually just a "latent vector" * In early GANs this was just a uniformly random vector --- ## Noise and Latent Codes * [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://arxiv.org/abs/1606.03657) was published way back in 2016 > In this paper, rather than using a single unstructured noise vector, we propose to decompose the input noise vector into two parts: (i) z, which is treated as source of incompressible noise; (ii) c, which we will call the latent code and will target the salient structured semantic features of the data distribution. --- ## Variations <div class="container"> <div class="col"> * There are a few ways to learn latent codes * Generally, gradient descent can compress information into a DNN * Is that memorization, or does it represent learning as we know it? </div> <div class="col"> <img style="width: 90%" class="r-stretch" src="./figures/UDL/Chap15/GanConditional.svg" /> <br/> <small>See <a href="https://udlbook.github.io/udlbook/">Understanding Deep Learning</a>, Figure 15.15.</small> </div> </div> --- ## Latent Code Impact <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/InfoGan_Figure2.png" /> <br/> <small>Read the <a href="https://arxiv.org/abs/1606.03657">InfoGAN paper</a>, see Figure 2.</small> </div> --- ## Meaning * By manipulating the latent code, we can force the generator to explore some dimension of the possible feature space * That sounds like we have a small program (the DNN) compressing a much larger space of images * In the 10 years since then, we have gotten better at controlling the exact meaning of the generator inputs * See the latest version of StyleGAN from 2022: [Third Time's the Charm? Image and Video Editing with StyleGAN3](https://arxiv.org/abs/2201.13433) --- ## Style GAN <div class="container"> <div class="col"> * The components of $w$ can be thought of as coarse, medium, and fine, depending upon where they influence the output * This leads to great control over the output images, as can be seem from example images in the [paper](https://openaccess.thecvf.com/content_CVPR_2019/html/Karras_A_Style-Based_Generator_Architecture_for_Generative_Adversarial_Networks_CVPR_2019_paper.html) </div> <div class="col"> <img style="width: 100%" class="r-stretch" src="./figures/UDL/Chap15/GanStyleGANArch.svg" /> <br/> <small>See the UDL text, Figure 15.19.</small> </div> </div> --- ## More Control * We can add more control over the latent code by using other methods * StyleGAN uses a vector, S, to insert a style into the synthetic image * [StyleCLIP: Text-Driven Manipulation of StyleGAN Imagery](https://openaccess.thecvf.com/content/ICCV2021/html/Patashnik_StyleCLIP_Text-Driven_Manipulation_of_StyleGAN_Imagery_ICCV_2021_paper.html), from 2021, showed that S could be manipulated using text prompts and CLIP * The goal is to get a new latent code, $w$, given a source code, $w_s$ --- ## Advances: Editing Directions * So optimize this: * $\underset{w \in \mathcal{W}+}{argmin}D_{CLIP}(G(w), t) + \lambda_{L2} ||w - w_s|| + \lambda_{ID} \mathcal{L}_{ID}(w)$ * $D_{CLIP}(G(w), t)$ is the cosine distance between the embedding of the Generator's output and the text embedding * ${L}_{ID}$ is a measure of output image similarity * Basically, find a new $w$ that changes the images a little as possible but brings the CLIP embedding as close as possible --- ## Generators Overview * So generators transform a latent variable into an output * The latent input has both semantic meaning and noise components * Let's say we had an output that didn't involve noise; what would that be? * Just a semantic meaning vector that produces a fixed output * Sounds like compression, right? * Let's move on to our last, and most explicit, example --- ## Puzzle Solving * In 2019, [On the Measure of Intelligence](https://arxiv.org/abs/1911.01547) suggested that "superhuman" performance of DNNs on various benchmarks was overselling progress * Humans didn't evolve to classify 224x244 pixel images and the DNNs get to train with millions of examples * AlphaGoZero played tens of millions of games, but a human playing 10 games a day for 100 years will only reach 36,000. * Is that really a proper comparison? * The author suggests testing the *acquisition* of new problem solving abilities --- > That is to say, intelligence is the rate at which a learner turns its experience and priors into new skills at valuable tasks that involve uncertainty and adaptation. > If an AI system has access to extensive, task-specific prior knowledge that is not available to a human, its performance on that task becomes a measure of the developer's cleverness in encoding that knowledge, not the AI's inherent intelligence. --- ## ARC * ARC is the Abstraction and Reasoning Corpus * It is designed measure *skill acquisition efficiency*, meaning how well an agent can learn to solve puzzles * Puzzles expect that an intelligence can infer the rule from a small number of examples * They are now on iteration three of the [ARC-AGI challenge](https://arcprize.org/arc-agi) --- ## Examples <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/ARC_fig8.png" /> <br/> <small>See Figure 8 of the <a href="https://arxiv.org/abs/1911.01547">ARC paper</a>.</small> </div> --- ## Examples <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/ARC_fig9.png" /> <br/> <small>See Figure 9 of the <a href="https://arxiv.org/abs/1911.01547">ARC paper</a>.</small> </div> --- ## Examples <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/ARC_fig10.png" /> <br/> <small>See Figure 9 of the <a href="https://arxiv.org/abs/1911.01547">ARC paper</a>.</small> </div> --- ## Types of Priors * What knowledge is being tested? * Ideas of shapes and objects being formed from sub-units * Object persistence in spite of noise or occlusion * Translation, rotation, scaling, etc * Counting, repetition * These are very general, and we know a DNN is generally capable of doing these tasks --- ## ARC Progress * The first version had 400 tasks in the training set and 400 in a public evaluation set * A hold out set of 100 tasks were used in competition from 2020 through 2024 * At the end of 2024, OpenAI used the ARC-AGI challenge as a demonstration of their latest reasoning model * It eventually scored 87% with high compute --- ## Cost of Progress * The cost on the x-axis is from the number of tokens consumed <div class="col"> <img style="width: 90%" class="r-stretch" src="./figures/arc-prize-leaderboard-arc1.png" /> <br/> <small>From the <a href="https://arcprize.org/leaderboard">ARC leaderboard</a>.</small> </div> --- ## Version 2 <div class="col"> <img style="width: 80%" class="r-stretch" src="./figures/arc-prize-leaderboard-arc2.png" /> <br/> <small>From the <a href="https://arcprize.org/leaderboard">ARC leaderboard</a>.</small> </div> --- ## Version 3 (new for 2026) <div class="col"> <img style="width: 90%" class="r-stretch" src="./figures/arc-prize-leaderboard-arc3.png" /> <br/> <small>From the <a href="https://arcprize.org/leaderboard">ARC leaderboard</a>.</small> </div> --- ## Comparison * Humans run at around 100 watts (all of us, not just the brainy parts) * Similar to a laptop, but we can also actuate things * Both susceptible to viruses * Could you pay a human to do these task? * Yeah, people would probably do [these](https://arcprize.org/tasks/tn36) for free --- ## Without Brute Force * Using one of those large reasoning models is overkill * It is like applying all of the world's text and images to solve a children's puzzle book * Is there a more efficient way to approach these problems? * Let's look at two papers * In [ARC-AGI Without Pretraining](https://arxiv.org/abs/2512.06104) the author's suggest that compressing the problems is a better approach * In [Less is More: Recursive Reasoning with Tiny Networks](https://arxiv.org/abs/2510.04871) the author outperforms several LLMs with lss than 0.01% of the parameters using a simplified hierarchical reasoning model <!-- Also: Tiny Recursive Reasoning with Mamba-2 Attention Hybrid https://arxiv.org/abs/2602.12078 --> --- ## CompressARC * [ARC-AGI Without Pretraining](https://arxiv.org/abs/2512.06104) attempt's to learn solutions to the ARC-AGI problems without *any* pretraining, using only the test puzzle itself for training > CompressARC tries to solve the problem of compressing the data into as short a program as possible, to obtain the puzzle solutions while keeping the program search feasible. In this case, the code must be entirely self-contained, receive no inputs, and must print out the entire ARC-AGI dataset of puzzles with any solutions filled in. --- ## Approach * Setup: * Puzzles are represented by tensors of shape [n_example, n_colors, width, height, 2] * Make a DNN classifier with outputs [n_examples, n_colors, width, height, 2] * The neural network must be equivariant to simple augmentations * Pick means and variance ($\mu$ and $\Sigma$) for $z$, the network input * Training: * Optimize network parameters $\theta$, $\mu$, and $\Sigma$ --- ## What is This? * The random input is acting like a latent input to a GAN or VAE * There is no noise in the output, so $z$ should become a compressed representation of the puzzle * But the reconstruction itself has errors * So the authors find the most common output as use that as the answer --- ## Visualization * The author's have a [website](https://iliao2345.github.io/blog_posts/arc_agi_without_pretraining/arc_agi_without_pretraining.html) with some visualizations <div class="col"> <img style="width: 90%" class="r-stretch" src="./figures/CompressARCSite_learning.png" /> <br/> <small>Compress ARC learning, as visualized by the authors.</small> </div> --- ## Learning and Compression * Each trained model is a generator for a particular type of puzzle * Training *is* the learning process * Occurs as puzzles are encountered rather than beforehand * So the compression, via random input and network parameters, is the learning --- ## Results * This solved about 20% of the puzzles * Which isn't bad, considering all it does is compression on each one individually * So there was no generalization and a limited number of samples --- ## Better Learning, Better AI * For the final set of course topics, we'll focus on this view of learning as compression * Knowledge is then a decompressor * In a biological context, this seems relevant * Being able to represent things efficiently means fewer calories and resources invested * So maybe this is the right way for us to go with artificial intelligence <!-- ARC-AGI Without Pretraining https://arxiv.org/pdf/2512.06104 https://iliao2345.github.io/blog_posts/arc_agi_without_pretraining/arc_agi_without_pretraining.html https://arcprize.org/arc-agi/2 The advisor has lots of interesting papers on sequence modeling and state spaces: https://scholar.google.com/citations?user=DVCHv1kAAAAJ&hl=en&oi=ao Basically, he wants to bring back recurrent networks that operate upon very long sequences. Others have combined the Mamba-2 recursive structure into tiny reasoning models --> <!-- Deep Spectral Methods: A Surprisingly Strong Baseline for Unsupervised Semantic Segmentation and Localization CVPR 2022 https://openaccess.thecvf.com/content/CVPR2022/html/Melas-Kyriazi_Deep_Spectral_Methods_A_Surprisingly_Strong_Baseline_for_Unsupervised_Semantic_CVPR_2022_paper.html https://github.com/lukemelas/deep-spectral-segmentation The spectral methods paper refers to: https://arxiv.org/abs/2109.14279 Localizing Objects with Self-Supervised Transformers and no Labels from 2021 --> <!-- Do we need a paper study each class? How about this one? https://www.mdpi.com/1424-8220/26/2/731 -->