State Space Model (SSM)

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State Space Model (SSM): A mathematical framework for representing dynamic systems using a set of state variables that capture the system's internal behavior over time. It describes how the system's state evolves in response to inputs and how the outputs are generated from the current state. SSMs are like maps for understanding how hidden systems work, even when you can't directly see their inner workings. They use a set of special variables, called state variables, to capture the system's current "state of being"—like a snapshot of its memory. These state variables act as clues, revealing how the system changes over time, responds to inputs, and produces outputs. It's like following a detective tracking the movements of suspects (state variables) based on clues (observations) and a set of rules (transition and observation equations).

Elements of a SSM:

  • State variables (x): Internal variables that represent the system's memory or current condition.
  • Inputs (u): External signals that influence the system's behavior.
  • Outputs (y): Observable quantities produced by the system.
  • State equations: First-order differential (continuous-time) or difference (discrete-time) equations that govern the evolution of state variables over time:
   - `dx/dt = Ax + Bu` (continuous-time)
   - `x(k+1) = Ax(k) + Bu(k)` (discrete-time)

- Output equations: Algebraic equations that relate the outputs to the state variables and inputs: 

   - `y = Cx + Du`

Generalizing Time-Domain:

  • Traditional time-domain representations (e.g., transfer functions) focus on input-output relationships.
  • SSMs go beyond this by explicitly representing the system's internal states, providing a deeper understanding of its internal dynamics.
  • This allows for modeling systems with:
    • Multiple inputs and outputs
    • Non-linearities
    • Time-varying parameters

Modeling and Analysis: 1. System Modeling: Representing physical, economic, biological, and other dynamic systems in a compact and flexible form. 2. Control System Design: Designing controllers for stabilization, tracking, regulation, and optimization. 3. Simulation: Simulating system behavior under various conditions to study its response and predict outcomes. 4. State Estimation: Using techniques like Kalman filters to estimate unknown states based on noisy measurements. 5. System Identification: Estimating model parameters from experimental data to create accurate system representations. 6. Fault Detection and Diagnosis: Monitoring system behavior to detect and diagnose faults or anomalies.

Applications:

  • Control engineering (e.g., robotics, aerospace, process control)
  • Economics (e.g., econometric modeling, time series analysis)
  • Signal processing (e.g., speech recognition, image processing)
  • Machine learning (e.g., recurrent neural networks, reinforcement learning)
  • Neuroscience (e.g., modeling brain dynamics)
  • And many more domains involving dynamic systems



State Space vs Transformer Models

We just did an episode on the first 90 days of Mamba literature. And one of the things that is really interesting about this new mechanism, the selective state space mechanism. Is that it does have different strengths and weaknesses compared to the Attention mechanism. Both in terms of how much memory it consumes where Attention mechanism is quadratic but the length of the input and that might be, by the way, one of the reasons like just as you talk about a torrent of data and a 1,000 samples per second, if that were to be naively translated to a 1,000 tokens per second. Then very quickly you're getting to a level of tokens that we have only very recently reached with frontier grade Transformers. It was only with GPT-4 a year ago that the public first got to see a quality 8,000 token Transformer. And before that it was like just a couple months where we had just seen the 4,000 before that as of like 18 months ago, 2,000 tokens was what you could really get from like the OpenAI API. So just the sheer volume of data may not limit itself super well to the Transformer but also another other When they break down these micro tasks. And look at what the Transformer can do and can't do one of the things that really struggles on is the hyper noisy environment there. There was a interesting result in this one. Mamba versus Transformer comparison paper, it's more about the state space mechanism and the Attention mechanism. Those are really the two things that are more dueling it out than the higher level architectures. And they're not even dueling it out because they actually work best together - spoiler. But in the super noisy environment where what actually matters is quite rare in what you're signaling, then the Transformer sometimes has a hard time converging and the intuition I've developed for that is because it's changing all the weights at the same time across like the entire range. Put it may be that the gradient is often dominated by noise and has a hard time converging on the signal. Whereas When I don't want to make everything about the selective state space model, so I do have an obsession about this as folks know. It is updating per token and so it seems like it has a more natural mechanism when the actual signal hits to say. Oh, and this is where I start to violate my anthropomorphizing policy but it has an ability to recognize when the signal hits and update in a more focused way on that one thing that really was supposed to matter, whereas the Transformer is updating everything all across, it's considering everything at once. And so, it seems like the signal can get lost in all that noise, the recurrent nature of the selective state space mechanism. ... to zero in and do the gradient on the signal when you have the signal and then of course there's still a lot of noise but that maybe can get separated from the signal because of this bit by bit level processing and updating. I'm not 100% just confident in that theory but it is consistent with all the evidence that I know of so far. Nick Labenz - The Cognitive Revolution

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