The Mapper Algorithm: Advancements and Applications in Topological Data Analysis

Topological Data Analysis (TDA) uses persistent homology and the Mapper algorithm to reveal hidden structures in high-dimensional data. Applications span finance for fraud detection, biomedicine for disease biomarkers, and genomics. Tools like GUDHI facilitate these insights.

Transcript

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You know, we spend so much time, I think, obsessing over the quantity of data, just the sheer volume, absolutely, it's all about big data. Yeah, we're constantly building bigger data lakes, training larger and larger models, just feeding this beast more and more numbers. It feels like we're trying to drink from a fire hose, right? Assuming that if we just swallow enough water, we'll eventually understand the ocean. That's a perfect analogy, and it's exactly the friction point we're hitting. We have built these AI systems that are, I mean, they're incredible at calculus, right? They're fantastic at finding the slope, the rate of change, the sort of immediate, local geometry of a line on a graph, but they're often just terrible at seeing the big picture. They miss the shape. They miss the shape. And when you miss the shape, you miss the meaning, you miss the context. And that is really the mission for this deep dive. We are looking at a whole stack of papers today. We've got stuff from Nature Communications frontiers and some really dense technical reports from places like super AGI and Yeshiva University, and they all claim that if you stop looking at the individual numbers and start looking at the shape of the data, the topology, you can predict things that theoretically should be impossible to predict things that feel like they should be random. Exactly. We're talking about spotting stock market crashes before the prices even start to drop or and this one blew my mind, identifying billion dollar fraud schemes just based on the geometry of how someone moves a mouse across a screen. It really does sound like science fiction or, you know, maybe something really esoteric and academic, but it's actually a very, very rigorous field called algebraic topology, and frankly, it is rapidly becoming the secret weapon for the next generation of AI. Why? What's the secret? Because it solves the one thing that deep learning, for all its power, really struggles with, and that structure, global structure. Okay, so let's ground this immediately, because when I hear topology, my brain just flashes back to, I don't know, high school math class, right? Or that old joke about how a topologist can't tell their coffee cup from their donut laughing, the classic, yes, because they both have one hole. It's a cliche, but it's actually the perfect starting point. Topology is the study of properties that don't change when you stretch or twist or bend an object, the hole in the donut is still there even if you squish it. That's what matters. Okay, I get that for a donut. But how on earth do you apply donut math to, say, a massive database of credit card transactions? Where's the hole? Well, you have to completely change how you visualize the data set. Right now, if I ask you to picture a data set, what do you see spreadsheet rows and columns? Or, if you're a data scientist, you might see a point cloud, right, just this huge swarm of individual dots floating in some high dimensional space, right, 1000s, millions of isolated data points. It's like a mist of numbers. And that's the problem. A mist has no structure. It's just stuff. The MDPI paper we looked at uses this fantastic analogy of an impressionist painting, okay, if you stand with your nose pressed up against a Seurat painting, all you see are individual dots of color. That's your raw data. It's just noise. It's complete noise. You can measure the distance between a blue dot and a red dot, but that tells you nothing about the picture of a woman holding a parasol. Yeah. TDA is the mathematical equivalent of taking three big steps back from the painting and squinting, and the dots blur together. They merge, they connect. Suddenly you see a face, a tree, the river, a coherent shape emerges from the noise. We call this process in TDA, filtration, filtration. Mathematically, what we're doing is we're expanding a little bubble radius around every single data point in that cloud. We call the radius epsilon. Okay, so every single dot gets its own growing bubble Exactly. And as those bubbles grow, they start to touch. They start to overlap. Suddenly, points that were isolated are now connected. An edge forms between them, and you aren't looking at 1000 isolated transaction records anymore. You're looking at a structure that's forming in real time right in front of you, a loop appears, a loop appears, or a void opens up, and that's where the analysis happens. It stops being about the individual points and the dots of paint, and it becomes all about the shapes they form. It's about the holes between them. So the insights are in the empty spaces. Precisely, we look for three specific things, and these are the sort of the building blocks of topology. The first is connected components. We call that homology group zero or H zero, H zero. And that's just clusters, groups of points that are all connected to each other. Yep. Think of it as finding the different islands in your data. The second thing is loops, or cycles. That's h1 these are literal one dimensional holes, like the hole in a donut. Okay, so clusters and loops and the third h2 are voids or cavities. These are two dimensional holes. Think of the empty space inside a basketball. It's a hollow sphere. So to bring it back to the real world, if I'm looking at my credit card data, and I see a big, tight cluster, an H zero component. What is that? That's probably. Normal behavior. It's a dense group of people buying coffee between eight and 9am or everyone buying groceries on a Saturday. It's a known, expected pattern, but if I see, I don't know, a weird loop forming an h1 feature, a loop could imply a cycle. Maybe it's a subscription service that bills every 30 days, or it could be something more sinister, a fraud ring that's cycling money between accounts. The loop itself is a feature that needs investigation and a void, an h2 hole that seems more abstract. A void is fascinating. It implies your data is wrapping around something. There's an empty space where you'd expect to see data, but there isn't any. Maybe it's a type of product that no one in a certain demographic ever buys. Or maybe it's a sophisticated fraudster who is carefully avoiding detection systems, creating a hollow space in the behavioral data, so the absence of data becomes the signal. Yes, and standard machine learning is terrible at this. It looks at geometry. It asks, Is this transaction far away from the average. It's a distance measurement. But fraudsters are smart. They know how to fake the average. They can make their transactions look close to normal, but they can't fake the topology. They can't easily fake the global shape. Their activity creates a different kind of structure, a weird, persistent hole that just doesn't belong. And that's the signal TDA can find. Okay, this is making sense, but I can't look at a 10,000 dimensional point cloud. This brings us to the mapper algorithm, which seems to be the real workhorse in this field. It is. It's the tool for simplification. It's our pair of glasses for looking at this stuff. The graphs in the show notes that were generated by mapper they look like networks, you know, nodes and edges, but they aren't just a standard network diagram. No, they're not. Mapper is a tool that lets you take that impossibly complex shape in high dimensions and project it down into a lower dimensional graph, something a human can actually look at. But, and this is the critical part, it preserves the connectivity, it keeps the essential shape. How does it do that? If you're squashing the data, don't you lose information. Isn't that? What always happens with dimensionality reduction? You do lose some information, but you preserve the most important topological information. And the trick is how it clusters the data. It uses overlapping bins. Overlapping bins. What does that mean? So think about sorting mail. Usually you put a letter in bin a or bin B. It can't be in both. That's standard clustering, right? Mapper is different. It projects the data onto a line, say, and it covers that line with intervals or bins that overlap with each other, so the single data point can fall into the end of Bin A and the beginning of Bin B simultaneously. Ah, so it creates a bridge between the two bins. You got it because that single plane exists in both bins. We know those two groups of data are related, so we draw a line connecting them. We connect the nodes in our graph. So the graph isn't showing every single data point, it's showing the relationships between clusters of points. Precisely, you end up with this topological network that shows you the backbone of your data. It reveals the flares, the loops, the tendrils and the dead ends that something like printable component analysis, which just flattens everything like a pancake, would completely obliterate you. Get a map of your data's shape. Okay, I think I've got the toolkit. We have the lens. We have the squinting. We have the map maker. Now I want to see this predict the future. Let's follow the money. Always a good place to start. The Super AGI report you mentioned talks about a $40 billion problem in online payment fraud. That's with a B. It's a staggering number, and the really terrifying stat in there is that global fraud losses are expected to surpass that number by 2027 it's a runaway train, and it's because the fraudsters are just evolving too quickly. The old way of doing things doesn't work, not at all. The old way was these static, rule based systems. You know, if the transaction is over$10,000 AMD, the IP address is in a different country, then lock it, which is so clumsy, you end up blocking legitimate users who are on vacation, and you completely miss the sophisticated thieves who use proxies, or, you know, break up their theft into hundreds of tiny transactions exactly they know the rules, so they play around them. The new frontier isn't analyzing the transaction amount, it's analyzing the behavior behind the transaction. This is where behavioral biometrics comes in. This stuff is fascinating. It's the shape of your physical interaction with the digital world. It's not what you type, it's how you type, the cadence, the rhythm, the speed. It's what angle you hold your phone at when you're logging in, how you move the mouse, yes, the curvature of your mouse movements, how fast you accelerate, whether you tend to overshoot and correct. All of these things create a signature that seems unbelievably subtle. Can it really be that unique? It is incredibly subtle, but it's also geometrically distinct. Think about it, even if a hacker steals your password and your username and logs into your bank account, they don't type like you. Their hands aren't my hands their nurse. System isn't your nervous system. They don't scroll like you. When they move the mouse from the username field to the password field, the path it takes forms a different topological shape in the data cloud than your path would. So the AI isn't just learning my password, it's building a topological model of me, my entire digital fingerprint in motion. That's it. It creates a point cloud of your normal behavior. It knows the shape of you. And then if a live session starts generating data points that deviate from that topology, if it creates a new weird loop or falls into a void where your data never, ever goes, it flags, it instantly, and the numbers in the source material back this up. It's not just a theory. No, this is in production and saving companies billions. The source mentions JPMorgan Chase reduced credit card fraud by 60% using these kinds of AI and machine learning techniques, 60% and Walmart, Walmart cut count takeover attempts by 60% as well. Super AGI even had a case study with an online marketplace where they reduced chargeback rates, which is a direct measure of fraud by 65% this is real. That's incredible. So that's protecting the downside. But what about what about the upside? What about predicting the stock market? This is where that research from Yeshiva University gets really heavy. That's his holy grail, right? It feels like it violates the efficient market hypothesis. If the signal is really there, shouldn't the market have already priced it in? You would absolutely think so. If you believe that markets are perfectly rational and all information is instantly available and processed, which we know they aren't. They are not, yes, and more importantly, TDA is looking at a layer of data that most trading algorithms, most quants, completely ignore. It's not looking at price or volume. It's looking at the complexity of the trading network, the shape of the market itself. They use this term that I had to look up, persistence, landscape norm. It sounds like a landscaping company for abstract math. Laughing. It does. But think of it as a a turbulence meter for the geometry of the market, a turbulence meter. I like that. The researchers, Goldman and gidia, analyzed the s, p5 100 in the period leading up to the big correction in early 2018 and they also looked at Bitcoin right before the crash in the spring of 2021 what were they looking for? Well, first they had to solve a big problem, which is finding the right time window to analyze. They call it the Goldilocks window. If your window is too short, all you see is random noise. If it's too long, you smooth everything out and you miss the signal. So how did they find the just right window? They used a tool called wavelets. These are like tiny, flexible waves that you can stretch and shift to match patterns and data. It's a very powerful signal processing technique. By using wavelets, they could find the perfect sliding window to analyze the market structure, second by second. Okay, so they found the window. They point their turbulence meter at the s, p5, 100. What did it show? It showed that before a crash, the market doesn't just, you know, go down, the underlying geometry of the market begins to change. They saw these incredibly sharp, dramatic spikes in this persistence landscape norm, the turbulence meter went into the red. Way into the red. The relationships between stocks, the correlation structures that normally hold the market together, they started to twist and fray and become topologically complex and unstable. So the structure is actually twisting under stress before it breaks. That is a perfect way to visualize it. The market becomes bumpy or rugged in a topological sense, this structural break, this geometric shift, happens before the massive sell off begins. That's the key. It's a pre signal. It's like hearing the ice start to crack and groan under your feet a full minute before you actually fall through the structural integrity of the market fails first, then the price plummets to catch up with that new reality that is a massive, massive edge. You're not just predicting price you're detecting the structural stress fracture in the economy before the bridge actually collapses. And because it's based on this abstract geometry, not on simple price momentum or moving averages. It's a signal that almost all traditional technical analysis just completely misses. It's a whole new dimension of information. Wow. So we can see the shape of a market crash. Let's shift from financial health to physical health. Can we see the shape of a heart attack? In a sense, yes. The MDPI source we looked at goes deep into diagnosing heart disease. And the numbers they're citing are, I mean, they're, frankly, hard to believe. At first glance, they're staggering. I saw the numbers the researchers Al janobi and Lee, they use the mapper algorithm on the famous Cleveland heart disease data set, and they achieve 99.32% accuracy in predicting the presence of heart disease. 99.32% I mean, that's that's practically perfect. That's better than most human experts. Why? Why is TDA so much better at this than a standard Doctor looking at an EKG or even a standard neural network? Because the heart is fundamentally a dynamical system. It's a pump. It has a rhythm, a cycle, if you plot the velocity of the heart muscle, again. Its position. We call this phase space reconstruction. A healthy heartbeat. Looks like a beautiful, clean loop, a circle, perfect circle, more or less it contracts, it relaxes, it returns to the start. It repeats. It has a very simple, elegant topology. It has one component and one loop and a diseased heart. It looks like a mess. It looks like a distorted scribble. The loop might not close properly. It might have extra little loops branching off of it, which represent arrhythmias or irregular beats. The clean topology is gone, and standard methods just don't see that. Standard methods look at things like average heart rate or peak voltage. Those are summary statistics. They're essentially taking that beautiful, complex loop and squashing it down into a single number. And when you squash the loop, you lose all the detail of the distortion, you lose the shape, you lose the shape exactly. TDA keeps the loop. It analyzes the shape of the cycle itself, and this is why it's so unbelievably effective for things like ventricular fibrillation or VF, that's the really dangerous one, right? Where the heart just kind of quivers instead of pumping. It's lethal if not treated immediately. And this is where the sources mentioned using TDA for AEDs, the automated defibrillators you see in airports and gyms, right? Those machines have to make a life or death decision in seconds. It has to decide shock or don't shock, yeah, and it has to be right. If it shocks a heart that has a normal rhythm, it can actually cause cardiac arrest. If it fails to shock a heart in VF, the person dies, the stakes could not be higher. So TDA gives it more certainty, a lot more the study by M Jihad and his team showed that TDA based features could distinguish shockable rhythms like VF from non shockable rhythms with 99.51% accuracy. That's the kind of reliability you need when a machine is literally holding the paddles to someone's chest. It's mission critical. But there was another aspect of cardiac health in the stack that I found just as interesting, maybe even more so, for the future of medicine, the patient's similarity idea, yes, this is research by a scientist named to Cody. It's not just about diagnosing one person as sick or healthy. It's about grouping people together in new ways. What did they do? They took echocardiogram data, ultrasounds of the heart from a large group of patients, and instead of grouping these people but the usual metrics like high blood pressure or age over 60, they use the mapper algorithm to group them by the topological shape of their heart's function. So they built a network of patients based on how their hearts were shaped, dynamically, exactly, and they found these hidden groups. The mapper algorithm revealed four distinct topological clusters in the patient population. And here's the kicker, okay, one of those clusters, which they called cluster four, looked pretty normal on paper. Their basic stats, ejection fraction, things like that, were very similar to the other groups. So a doctor looking at a spreadsheet wouldn't see anything particularly alarming, probably not. But topologically, the shape of their heart function was distinct, and the patients in that topologically distinct cluster, four had double the risk of major adverse cardiovascular events, heart attacks, strokes, death, double the risk, and it was invisible to standard metrics, invisible the TDA graph just screamed danger for this group. And this is where it gets so powerful for personalized medicine, you're no longer just a 50 year old male with slightly high cholesterol, they're a cluster four topology, right? Which tells your doctor that you need a much more aggressive preventative strategy than someone whose heart has a cluster one topology. It allows us to stratify risk in a way that linear statistics just can't handle. Okay, that's profound. Let's pivot from the pump that powers the body to the processor that runs it, the brain. My favorite topic, the neuroscience applications here are fascinating because they seem to fundamentally challenge the old almost phrenology, like idea that intelligence is just about having a big frontal lobe or a heavier brain, right? The bigger is better theory. It's a very simplistic idea, and it just doesn't hold up to the data. We've known for a while that brain size doesn't correlate perfectly with IQ. This new study by Wilcox and Bardy suggests that intelligence isn't about size or even activity in one specific smart region. It's about efficiency and the overall network architecture. It's a topological property. So they map the brains of over 800 young adults. This was from the Human Connectome Project, right? Yes, an amazing data set. They combined structural MRI, which shows the physical wiring of the brain, with functional MRI, which shows which areas are active when someone is performing a task, and they found that smart brains are, for lack of a better word, wired differently. They're wired more like efficient social networks or maybe a really well designed subway system. How so they use this term from sociology, weak ties. Now, in sociology, a weak tie is like an acquaintance, someone you don't know that well, but who connects you to a completely different social circle than your best friends do exactly your strong ties are your best friends. You all live in the same neighborhood. You know the same people, you hear the same. Gossip. It's a very dense, high energy, but ultimately redundant network. You don't get new information, not really. Weak ties are the bridges. They're the acquaintance who tells you about a job opening in a totally different industry. In the brain, these weak ties are the long range physical connections that link different neural modules together, say the visual cortex to the auditory cortex. They're metabolically low energy, but they are incredibly high efficiency bridges for information. So a smart brain isn't just constantly blasting energy through these thick super highway cables. It's using these, these efficient, delicate back roads to jump from one brain region to another. It's an energy optimization problem. At its core, the brain uses 20% of our body's energy. It has to be efficient. The study found that more intelligent brains rely more heavily on these weak ties to switch states, quickly and with low energy cost, switch states. What do you mean, like switching from a state of daydreaming to a state of doing calculus? Those are two vastly different configurations of your entire brain's network activity. The weak ties are what facilitate that global shift. They reconfigure the network on the fly, and this ties directly into the other big concept they found the small world network idea. Yes, this is a classic concept in network theory. A Small World network has two key features. First, it has high clustering, which means your local neighbors are also neighbors with each other. That's your dense local processing, your strong ties, okay, but it also has very short path lengths, which means you can get from any node in the entire network to any other node in just a few jumps. That's your global efficiency. Your weak ties provide those shortcuts. So the visual cortex talks to itself a lot. That's the high clustering. But it can also ping the memory center instantly if it needs to. That's the short path length. Precisely. It's the best of both worlds. It prevents information from getting trapped in local clicks. And TDA gives us the mathematical tools to actually measure this small worldness of a brain. They also identified another key feature they called modal control. Modal control. Think of these as the brain's gear shifters. There are specific hub regions that have the unique ability to push or drive the entire brain network into states that are difficult to reach, like solving a really complex, novel problem, exactly a state that requires a very specific, unusual configuration of brain activity. The study found that high intelligence correlates with having better gear shifters regions that can drive the network into these complex problem solving modes on demand. So intelligence is basically a topological property. It's the shape of your brain's network, a shape that balances local processing with global reach, all while minimizing the energy it takes to think. It's the shape of thought itself that is a beautiful way to put it. We're really moving from mapping the static anatomy of the brain to mapping the dynamic topology of thought. All right, we've covered the body, the brain and the wallet. Now let's talk about the machine itself, because for all the incredible hype, AI has a massive black box problem. It's the single biggest hurdle for adoption in high stakes fields like medicine or finance. If an AI denies your loan application or flags a spot in your MRI as cancerous, you have a right to know why. You can't just accept the answer. The neural net said so, and usually, if you try to look inside a deep neural network, it's just it's a matrix of millions of numbers. It's completely meaningless to a human. How does TDA help us open that box by visualizing the data's journey. A neural network is essentially a transformation machine. It takes your input data, say the pixels of an image, and it stretches and warps it layer by layer until the data for cats is cleanly separated from the data for dogs. So it's a geometric transformation. It's a topological one, really, yeah, and we can use TDA to map that warping process. There was a tool mentioned in the sources called topo map X. It uses the mapper algorithm to visualize how the neural network routes data through its internal layers. So you can actually see the path the data takes. You can see the decision boundaries it's creating, you might see that the network has created a specific topological cluster for all the rejected loan applications. And if you then analyze that cluster and realize that all the people in it live in the same three zip codes, you've just found a huge bias in your model. Exactly. You can spot topological bias. You can also do things like calculate the topological entropy of the network's activations, which is a measure of its internal complexity. What would that tell you? If the entropy is too high, the model might be confused or over fitting to the training data. If it's too low, it might be oversimplifying the problem. It's like giving the AI's brain an MRI to see if it's healthy and working correctly. It makes it explainable. Okay, now we cannot finish the segment without talking about the paper with hands down the wildest title in the entire stack. I know which way. Do you mean topological zigzag spaghetti? It is a mouthful, isn't it? It sounds completely ridiculous, like a bad menu item at an Italian restaurant. But the problem. It solves is actually the biggest bottleneck in TDA, isn't it? Everything we've discussed so far, the stock market snapshots, the brain scans, the heartbeats, those are all static pictures. That is the fundamental limitation of classic TDA. It takes a photo, but the world is a movie. Molecules are constantly vibrating. Social networks are evolving. Financial markets are fluctuating in real time. If you just take a snapshot, you miss the motion, you miss the change over time. So zigzag persistence is the video camera for topology. That's a great way to think about it. Standard persistence, homology tracks when a hole is born and when it dies in a single, static data set. Zigzag persistence tracks how a topological future, a hole, a loop persists, disappears and reappears across a sequence of changing data sets. So if you have a social network and friendships are forming and breaking, it can track a hole in the network as it changes shape over time, exactly it zigs and zags through time and the spaghetti. Please explain the spaghetti. The spaghetti is the clever new visualization tool they developed to summarize this complex time varying information. It looks like a tangle of spaghetti strands, where each strand represents a topological feature moving through time and across different scales. It's a way to capture all of that dynamic information in one picture. But the killer app here isn't making pasta diagrams. It's for generative AI, specifically for diffusion models. These are the models that are powering the revolution in generative AI, things like mid journey or dolly for images, but also, and maybe more importantly, the new wave of AI for drug discovery and material science, right? Because generating a new drug. Molecule isn't a single step. It's a process. It starts as random noise and slowly resolves into a coherent structure. It's an evolution. It's a process of denoising, and if the AI doesn't understand the topology of that evolutionary process, it can easily generate a molecule that looks okay in a 2d drawing, but is physically impossible to construct in 3d or is chemically unstable. It builds a beautiful, but useless molecule, exactly zigzag spaghetti. Allows the AI to understand and maintain the correct topology during the generation process itself. It guides the AI to create outputs that are not just plausible, but structurally sound and robust. It basically stabilizes the AI's hallucinations, that's a perfect way to put it. In their experiments, using the zigzag approach, reduced the variability in the generated graphs by a factor of five. That means the AI is five times more consistent and reliable in generating valid, useful structures. If we want AI to design the next cure for cancer or the next super material. We absolutely need that kind of topological stability. We're almost at the end here, but we have to touch on the cost. I mean, calculating all these bubbles and triangles and tetrahedrons and holes, and it sounds punishingly expensive. It is. It's computationally heavy. As the number of data points and the dimensions grow the number of possible triangles and tetrahedrons you have to check for just explodes. It's a combinatorial nightmare. It is, frankly, the main reason TDA hasn't completely taken over the world yet, but the stack, our stack of sources, it mentions a solution that's coming over the horizon, and that's quantum computing. That is the end game. The ijert source goes into this emerging field of quantum TDA. How does a quantum computer help you count holes in data. Well, quantum computers are inherently designed to search vast complex bases all at once. They mentioned using Grover optimized algorithms. Grover's algorithm is a famous quantum search method that provides a quadratic speed up for unstructured search problems. A quadratic speed up it can drastically accelerate the process of finding cliques, which are groups of fully connected points, and finding those cliques is the very first and often most expensive step in building the simplicial complexes that TDA analyzes. So a task that might take a classical supercomputer weeks to map the topology of a complex protein. A quantum computer could do it in minutes, hours, potentially, we are in the very early days, but we're already seeing hybrid quantum classical experiments using IBM's quantum processors to analyze the topology of small biomolecules. We are not there yet for massive big data problems, but the theoretical path is clear. Quantum computers will eventually build the complex maps that TDA needs to read. It's really amazing to pull all of these threads together, whether it's the weak ties in an intelligent brain, or the geometric turbulence in a crashing market or the shape of a fraudulent mouse movement. The recurring theme is that the raw numbers we've been obsessed with are just shadows. The reality is the structure. The structure is the signal. For decades, we've been so focused on collecting all the dots that We completely forgot to look at how they connect to each other, and we missed the picture. There was one last concept, and it was buried deep in that super AGR report on fraud that I want to leave people with to think about federated learning. This is the privacy breakthrough that could change everything. It connects perfectly to TDA. If the shape of the data is what holds the truth, then I don't actually need to see your private, sensitive medical records to help cure cancer. I just need to see the shape of them right in traditional federated learning, you train a model locally on private data, and then you share the model's updated weights, which can sometimes still leak information, but TDA offers a different, more powerful path. The hospital keeps all the patient data locked down. It never leaves their server never they run the TDA algorithm locally and generate the persistence diagram, the topological barcode of their patient data's shape, and all they send to the Central Research Hub is that barcode, and that bar code contains no names, no social security numbers, no specific lab values. It's just an abstract summary of the geometry of their data set. Exactly. It's fundamentally anonymous, but I can take that barcode and combine it with barcodes from 100 other hospitals around the world to build a global topological model of human health or of a specific disease. It preserves privacy perfectly while sharing the structural wisdom hidden in the data. It's a powerful thought. If the shape of data really is the shape of reality, are we just now for the very first time learning how to see the world in high definition? I think we're finally putting on the 3d classes, and once you see the depth, you can't really go back to looking at the flat world again. On that note, thank you for diving deep with us today. Next time you're looking at a spreadsheet full of numbers, try to take a step back and squint. You might just see a landscape for a donut. Thanks for listening. If you want to see the shape of your own data or just figure out what a persistence barcode actually looks like, we'll have links in the show notes to the mapper algorithm and some of the key papers we discussed. And of course, don't forget to subscribe for the next deep dive. See you next time.