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We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data?...

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Mathelirium

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We often hear that Machine Learning models learn patterns in data. But what does that actually look like in Geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a Self-Organizing Map (SOM), a simple unsupervised neural model, learn the shape of a 3D datasets l, one static and the other dynamic. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud. It knows nothing about the structure underneath. Learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. Python code is available for Subscribers. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML
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We often hear that Machine Learning models learn patterns in data. But what does that actually look like in Geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a Self-Organizing Map (SOM), a simple unsupervised neural model, learn the shape of a 3D datasets l, one static and the other dynamic. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud. It knows nothing about the structure underneath. Learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. Python code is available for Subscribers. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

Mathelirium

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A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982
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A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982

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The Machine That Learns The Law Behind The Data A very very interesting US Patent US10963540B2 - Physics Informed Learning Machine describes a learning system that does not begin with data alone. It begins with a physical model, usually written as a differential equation (or PDE) dx/dt = f(x,t) A normal Machine Learning model sees scattered data and tries to fit it. A physics-informed learning machine starts with a law. Then it treats the data as evidence that updates what the model believes about the physical system. For this application, I use the patent idea on NASA C-MAPSS Turbofan engine data. The machine watches multivariate telemetry from a degrading engine and infers a hidden health state that is not measured directly. From that posterior belief, it estimates the engine’s remaining useful life. In the main 3D scene, the engine lifetime is turned into a tunnel. The spiral ribbons are real sensor channels evolving over cycle-time. The glowing core is the inferred health state. The surrounding cloud is uncertainty. The orange wall ahead is the predicted failure horizon. So the big picture is: sensor evidence comes in, posterior belief tightens, and the machine moves from uncertainty toward a concrete failure prediction. The inset posteriors make that explicit. The health posterior shows where the model believes the hidden engine condition sits at the current moment, and how sharply it believes it. The RUL posterior shows the same idea for remaining life... early on it is broad, later it shifts left and narrows as the machine becomes more certain about how close failure is. This idea is not limited to engines. The same idea can apply to data centers, CPUs, GPUs, cooling systems, power grids, robotics, batteries, and any machine that produces telemetry while obeying physical constraints. In an age where machine learning runs on massive hardware infrastructure, this kind of model matters: it can turn noisy sensor streams into early warnings before expensive systems fail.
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The Machine That Learns The Law Behind The Data A very very interesting US Patent US10963540B2 - Physics Informed Learning Machine describes a learning system that does not begin with data alone. It begins with a physical model, usually written as a differential equation (or PDE) dx/dt = f(x,t) A normal Machine Learning model sees scattered data and tries to fit it. A physics-informed learning machine starts with a law. Then it treats the data as evidence that updates what the model believes about the physical system. For this application, I use the patent idea on NASA C-MAPSS Turbofan engine data. The machine watches multivariate telemetry from a degrading engine and infers a hidden health state that is not measured directly. From that posterior belief, it estimates the engine’s remaining useful life. In the main 3D scene, the engine lifetime is turned into a tunnel. The spiral ribbons are real sensor channels evolving over cycle-time. The glowing core is the inferred health state. The surrounding cloud is uncertainty. The orange wall ahead is the predicted failure horizon. So the big picture is: sensor evidence comes in, posterior belief tightens, and the machine moves from uncertainty toward a concrete failure prediction. The inset posteriors make that explicit. The health posterior shows where the model believes the hidden engine condition sits at the current moment, and how sharply it believes it. The RUL posterior shows the same idea for remaining life... early on it is broad, later it shifts left and narrows as the machine becomes more certain about how close failure is. This idea is not limited to engines. The same idea can apply to data centers, CPUs, GPUs, cooling systems, power grids, robotics, batteries, and any machine that produces telemetry while obeying physical constraints. In an age where machine learning runs on massive hardware infrastructure, this kind of model matters: it can turn noisy sensor streams into early warnings before expensive systems fail.

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A Neural Network Can Grow New Neurons Where It Is Confused? In 1994, Bernd Fritzke published A Growing Neural Gas Network Learns Topologies. He introduced a network that starts small, follows incoming data, and inserts new neurons where its error is highest. In the animation, the fog is the drifting data. The glowing nodes are neurons. The fibers are learned connections. The network grows into a living skeleton of the manifold.
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A Neural Network Can Grow New Neurons Where It Is Confused? In 1994, Bernd Fritzke published A Growing Neural Gas Network Learns Topologies. He introduced a network that starts small, follows incoming data, and inserts new neurons where its error is highest. In the animation, the fog is the drifting data. The glowing nodes are neurons. The fibers are learned connections. The network grows into a living skeleton of the manifold.

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Individual Neuron: Neural Network A neuron in a neural network performs a weighted sum of inputs, adds a bias, and applies an activation function like sigmoid, ReLU, or tanh, introducing non-linearity. This output helps the neuron learn and represent patterns in the data.
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Individual Neuron: Neural Network A neuron in a neural network performs a weighted sum of inputs, adds a bias, and applies an activation function like sigmoid, ReLU, or tanh, introducing non-linearity. This output helps the neuron learn and represent patterns in the data.

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“But I saved the history on CD, DVD, Magnetics” That data will all decay in a MTBF of +\-50 years. The cloud is deleting as much data that is being produced daily. At some point in the future we would have lost a majority of the 2000s-2040s. We are the amnesia generation.
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“But I saved the history on CD, DVD, Magnetics” That data will all decay in a MTBF of +\-50 years. The cloud is deleting as much data that is being produced daily. At some point in the future we would have lost a majority of the 2000s-2040s. We are the amnesia generation.

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Geometry of Machine Learning Models - Gaussian Process Kernel In 1948 Norbert Wiener framed prediction as a correlation problem, and in the 1970s George Wahba clarified that picking a smoothness preference is the same as picking a kernel. The motivation is that whenever data i s sparse and noisy, you don't just want a fit, you want a fit with calibrated uncertainty.
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Geometry of Machine Learning Models - Gaussian Process Kernel In 1948 Norbert Wiener framed prediction as a correlation problem, and in the 1970s George Wahba clarified that picking a smoothness preference is the same as picking a kernel. The motivation is that whenever data i s sparse and noisy, you don't just want a fit, you want a fit with calibrated uncertainty.

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Peter Ndegwa: In a crisis like we faced last year, when there were the Gen Z protests, there was an internet issue and the issue of data. It doesn’t matter what story we say, at the end of the day, we pride ourselves on delivering and always being safe secure. In a way, we disappointed customers because the internet was not working at that point. On the Gen Z side, it’s a learning for everyone, it’s a learning from a political and from a corporate perspective. People still trust our brand, our reputation is still very strong #CitizenExplainer Yvonne Okwara

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Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.
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Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

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There's a fruit fly walking around right now that was never born. eon just released a video where they took a real fly's connectome — the wiring diagram of its brain — and simulated it. Dropped it into a virtual body. It started walking. Grooming. Feeding. Doing what flies do. Nobody taught it to walk. No training data, no gradient descent toward fly-like behavior. This is the opposite of how AI works. They rebuilt the mind from the inside, neuron by neuron, and behavior just... emerged. It's the first time a biological organism has been recreated not by modeling what it does, but by modeling what it is. A human brain is 6 OOM more neurons. That's a scaling problem, something we've gotten very good at solving. So what happens when we have a working copy of the human mind?
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There's a fruit fly walking around right now that was never born. eon just released a video where they took a real fly's connectome — the wiring diagram of its brain — and simulated it. Dropped it into a virtual body. It started walking. Grooming. Feeding. Doing what flies do. Nobody taught it to walk. No training data, no gradient descent toward fly-like behavior. This is the opposite of how AI works. They rebuilt the mind from the inside, neuron by neuron, and behavior just... emerged. It's the first time a biological organism has been recreated not by modeling what it does, but by modeling what it is. A human brain is 6 OOM more neurons. That's a scaling problem, something we've gotten very good at solving. So what happens when we have a working copy of the human mind?

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New PNAS paper. Historical GDP per capita data is scarce, but data on the places of birth, death, and occupations of famous individuals is abundant. In this paper we estimate the historical GDP per capita of hundreds of regions in Europe and North America using a machine learning model that leveraged data on about 500k famous biographies. Our estimates more-or-less quadruple the availability of historical GDP per capita estimates for the last 700 years. So why use biographies to augment historical GDP per capita data? Biographical data contains information about people who might have contributed directly to economic growth, like James Watt, or that were attracted to wealthy places looking for patrons, like Michelangelo. So we--mainly Philipp (Philipp Koch)--used this data to construct hundreds of features describing each European region. Then, we trained a machine learning model to find the features that explained most of the variance in a cross-validation test, where we split regions multiple times into a training set and a test set. On average, the model explained about 90% of the variance in GDP per capita of the regions it had not seen during training. But we wanted to go further, and Philipp really went to town by looking at different ways to validate our estimates. We found our estimates correlate positively with historical measures of wellbeing, church building activity, urbanization, and body height. We also used these measures to reproduce the basic Atlantic trade result of Acemoglu, Johnson, and Robison and to explore the economic consequences of the famous Lisbon earthquake of 1755. But what I personally loved most about this project, other than working with Philipp Koch and V, is that it shows that we can use machine learning methods not only to explore the future, but the past. There is a bright and growing future in the use of machine learning for economic history. Hope you enjoy the paper and the data. You can find links to the paper and a data exploration tool in the first comment.
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New PNAS paper. Historical GDP per capita data is scarce, but data on the places of birth, death, and occupations of famous individuals is abundant. In this paper we estimate the historical GDP per capita of hundreds of regions in Europe and North America using a machine learning model that leveraged data on about 500k famous biographies. Our estimates more-or-less quadruple the availability of historical GDP per capita estimates for the last 700 years. So why use biographies to augment historical GDP per capita data? Biographical data contains information about people who might have contributed directly to economic growth, like James Watt, or that were attracted to wealthy places looking for patrons, like Michelangelo. So we--mainly Philipp (Philipp Koch)--used this data to construct hundreds of features describing each European region. Then, we trained a machine learning model to find the features that explained most of the variance in a cross-validation test, where we split regions multiple times into a training set and a test set. On average, the model explained about 90% of the variance in GDP per capita of the regions it had not seen during training. But we wanted to go further, and Philipp really went to town by looking at different ways to validate our estimates. We found our estimates correlate positively with historical measures of wellbeing, church building activity, urbanization, and body height. We also used these measures to reproduce the basic Atlantic trade result of Acemoglu, Johnson, and Robison and to explore the economic consequences of the famous Lisbon earthquake of 1755. But what I personally loved most about this project, other than working with Philipp Koch and V, is that it shows that we can use machine learning methods not only to explore the future, but the past. There is a bright and growing future in the use of machine learning for economic history. Hope you enjoy the paper and the data. You can find links to the paper and a data exploration tool in the first comment.

César A. Hidalgo

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I don’t know if we live in a Matrix, but I know for sure that robots will spend most of their lives in simulation. Let machines train machines. I’m excited to introduce DexMimicGen, a massive-scale synthetic data generator that enables a humanoid robot to learn complex skills from only a handful of human demonstrations. Yes, as few as 5! DexMimicGen addresses the biggest pain point in robotics: where do we get data? Unlike with LLMs, where vast amounts of texts are readily available, you cannot simply download motor control signals from the internet. So researchers teleoperate the robots to collect motion data via XR headsets. They have to repeat the same skill over and over and over again, because neural nets are data hungry. This is a very slow and uncomfortable process. At NVIDIA, we believe the majority of high-quality tokens for robot foundation models will come from simulation. What DexMimicGen does is to trade GPU compute time for human time. It takes one motion trajectory from human, and multiplies into 1000s of new trajectories. A robot brain trained on this augmented dataset will generalize far better in the real world. Think of DexMimicGen as a learning signal amplifier. It maps a small dataset to a large (de facto infinite) dataset, using physics simulation in the loop. In this way, we free humans from babysitting the bots all day. The future of robot data is generative. The future of the entire robot learning pipeline will also be generative. 🧵
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I don’t know if we live in a Matrix, but I know for sure that robots will spend most of their lives in simulation. Let machines train machines. I’m excited to introduce DexMimicGen, a massive-scale synthetic data generator that enables a humanoid robot to learn complex skills from only a handful of human demonstrations. Yes, as few as 5! DexMimicGen addresses the biggest pain point in robotics: where do we get data? Unlike with LLMs, where vast amounts of texts are readily available, you cannot simply download motor control signals from the internet. So researchers teleoperate the robots to collect motion data via XR headsets. They have to repeat the same skill over and over and over again, because neural nets are data hungry. This is a very slow and uncomfortable process. At NVIDIA, we believe the majority of high-quality tokens for robot foundation models will come from simulation. What DexMimicGen does is to trade GPU compute time for human time. It takes one motion trajectory from human, and multiplies into 1000s of new trajectories. A robot brain trained on this augmented dataset will generalize far better in the real world. Think of DexMimicGen as a learning signal amplifier. It maps a small dataset to a large (de facto infinite) dataset, using physics simulation in the loop. In this way, we free humans from babysitting the bots all day. The future of robot data is generative. The future of the entire robot learning pipeline will also be generative. 🧵

Jim Fan

165,215 Aufrufe • vor 1 Jahr

The solar system is a study in chronological lag. When you look at the Sun, you are peering 8 minutes and 13 seconds into the past. By the time that same light reaches Neptune, over four hours have vanished. ​We tend to visualize space as a static map, but it is actually a staggered broadcast of ancient photons. This visualization captures the profound isolation of the outer giants. While Mercury is practically bathing in real time, the rest of the family is drifting in a massive delay. We are all orbiting the same flame, just at different points in history. Credit: cosmicverse
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The solar system is a study in chronological lag. When you look at the Sun, you are peering 8 minutes and 13 seconds into the past. By the time that same light reaches Neptune, over four hours have vanished. ​We tend to visualize space as a static map, but it is actually a staggered broadcast of ancient photons. This visualization captures the profound isolation of the outer giants. While Mercury is practically bathing in real time, the rest of the family is drifting in a massive delay. We are all orbiting the same flame, just at different points in history. Credit: cosmicverse

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This is the best way to understand how ML models actually work! Use Drawdata to draw a 2D dataset in Jupyter. Use it to actively pick data from the widget and update the model as the data is being drawn! Fully interactive, real-time, and open-source!
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This is the best way to understand how ML models actually work! Use Drawdata to draw a 2D dataset in Jupyter. Use it to actively pick data from the widget and update the model as the data is being drawn! Fully interactive, real-time, and open-source!

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Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
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Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

106,981 Aufrufe • vor 5 Monaten

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.
0:20

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

45,688 Aufrufe • vor 2 Monaten

‘New Russia-Ukraine Talks May Take Place in Geneva’ – Kellogg ‘When President Trump spoke to Putin a little over a week ago, the Russians said they would present what they called a “memorandum.” I call it a “term sheet.” What does that mean? It’s a document that spells out how to achieve peace. The Ukrainian side has already given their version. Now we need to get the same from Russia. And then we put the two documents together, see how we can combine them, determine what is acceptable, what is unacceptable, and where we can find common ground. And when that is ready, there will be another meeting. We think, perhaps, in Geneva. We would like to hold it in the Vatican, and we had everything ready. But the Russians refused to go to the Vatican. So Geneva is probably the next place. And then we can organize a meeting of the three leaders: the President of the United States, Putin, and Zelenskyy — and conclude some document that will end this war.’ - SK
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‘New Russia-Ukraine Talks May Take Place in Geneva’ – Kellogg ‘When President Trump spoke to Putin a little over a week ago, the Russians said they would present what they called a “memorandum.” I call it a “term sheet.” What does that mean? It’s a document that spells out how to achieve peace. The Ukrainian side has already given their version. Now we need to get the same from Russia. And then we put the two documents together, see how we can combine them, determine what is acceptable, what is unacceptable, and where we can find common ground. And when that is ready, there will be another meeting. We think, perhaps, in Geneva. We would like to hold it in the Vatican, and we had everything ready. But the Russians refused to go to the Vatican. So Geneva is probably the next place. And then we can organize a meeting of the three leaders: the President of the United States, Putin, and Zelenskyy — and conclude some document that will end this war.’ - SK

Zlatti71

30,186 Aufrufe • vor 1 Jahr

"We're not a consumer company, we're the shortest path to artificial labor." - Bernt Bornich, founder of 1X "It just happens to be that the shortest path is through consumer, because you need to live and learn among people." "Model intelligence comes from the diversity of your AI dataset. There isn't enough variance (if you are just) in a factory." "We want to deploy robots there (in homes), where we get data through a truly intelligent machine that doesn't just repeat a motion."
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"We're not a consumer company, we're the shortest path to artificial labor." - Bernt Bornich, founder of 1X "It just happens to be that the shortest path is through consumer, because you need to live and learn among people." "Model intelligence comes from the diversity of your AI dataset. There isn't enough variance (if you are just) in a factory." "We want to deploy robots there (in homes), where we get data through a truly intelligent machine that doesn't just repeat a motion."

TBPN

15,742 Aufrufe • vor 1 Jahr