Compressed Sensing in Venture Capital
What's the mathematics behind compressed sensing? Why is it relevant to venture capital?
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I had worked on compressed sensing as an undergrad. And had tucked it away for a long time. It recently came back to me and the underlying concept made me compare it to the structure of venture capital. More specifically, the job of venture capitalists. So I decided to write about it.
The Mathematics of Compressed Sensing
In traditional signal processing, capturing an entire dataset is often considered necessary to reconstruct a high-fidelity signal. But this assumption becomes impractical when acquiring full data is expensive, slow, or physically impossible. This is where compressed sensing comes into play. It’s a mathematical framework that challenges this paradigm.
In essence, compressed sensing says that a sparse signal can be reconstructed from far fewer measurements than previously thought necessary.
At its core, compressed sensing relies on two fundamental principles:
Sparsity: It refers to the idea that most signals have only a small number of nonzero coefficients.
Incoherence: This ensures that the way we sample the signal spreads information evenly across different dimensions, preventing loss of critical details.
Mathematically, compressed sensing leverages convex optimization techniques to recover signals from incomplete data. How? By finding the sparsest possible solution that fits the observed measurements. This approach is particularly powerful in areas like medical imaging, astronomy, and remote sensing. Why? Because acquiring complete datasets is prohibitively expensive or infeasible.
The implications are profound: we don’t need to observe everything to understand everything. By exploiting underlying structure, we can infer the whole from the part. And we can reconstruct meaningful signals from seemingly inadequate data.
Venture Capital as an Inference Problem
Just like a signal in the Compressed Sensing framework, a startup’s potential is a high-dimensional entity that cannot be fully measured at the moment of investment. Venture capitalists face a difficult task: they must predict whether a fledgling startup will thrive based on a sparse and noisy set of observations. We get to see fragments of the full picture scattered with noise and ambiguity.
Late stage VCs and public market investors can follow traditional data-driven approach akin to Nyquist-Shannon sampling where you collect exhaustive data before making a decision. But early stage VCs don’t have that luxury.
By the time all data is available, the opportunity has either been seized by another investor or the startup has perished from lack of capital. So investors must learn to reconstruct the underlying truth from a few well-chosen measurements.
Identifying Structure in Incomplete Information
Both venture capital and compressed sensing hinge on identifying the structure within incomplete information. In compressed sensing, sparsity assumptions allow for reconstruction from limited samples. In VC, experienced investors develop mental models and heuristics that act as priors. And these priors guide them toward startups with hidden potential.
For example, let’s say a startup’s revenue is modest. A good VC might recognize a pattern such as an unusually high rate of customer engagement or a founding team with prior successful exits. These sparse but high-impact features serve as proxy signals for future success. Just as an underdetermined system of equations can be solved when the solution is sparse, an investor can make accurate inferences when they identify startups that fit known success patterns.
The Role of Priors and Inference
In compressed sensing, priors in the form of sparsity constraints enable signal reconstruction. In venture capital, priors come in the form of intuition + experience + historical data.
Investors don’t just look at isolated data points. They connect them in ways that reveal latent structure.
In early-stage investing, financial statements are often too limited to be conclusive. So investors may prioritize signals such as founder-market fit—a startup’s alignment with a problem space based on the team’s background and expertise. This is akin to how compressed sensing relies on sparsity as a constraint. This allows for reconstruction from limited measurements rather than exhaustive sampling.
Inference in venture capital mirrors Bayesian updating where new information refines an initial belief.
Just as compressed sensing refines an estimate as more measurements are taken, an investor continuously adjusts their understanding as they gather new data.
When Reconstruction Fails: Noise and Misinterpretation
Not all signals can be reconstructed accurately. And not all startups that receive investment succeed. Just as compressed sensing fails when a signal lacks sufficient sparsity or when noise overwhelms the measurement process, VCs can misjudge a startup’s potential when key signals are misleading.
For example, hype cycles in technology can generate false positives. These are startups that appear promising due to superficial traction but lack the structural integrity to scale. On the flip side, some of the best investments are false negatives in traditional evaluation metrics i.e. companies that appear unimpressive but contain hidden attributes that only a trained eye can recognize.
The Art and Science of Incomplete Data
Compressed Sensing is a triumph of mathematical insight. It allows us to extract deep truths from sparse observations by leveraging structure and priors. Venture Capital is an exercise in pattern recognition and inference. Investors must make high-stakes decisions under uncertainty, reconstructing the future potential of companies from incomplete and noisy data.
But at the same time, venture capitalists are in the business of outliers. In true power law fashion, extreme outliers are the ones that do all the work for you. So how do spot these outliers?
In both domains, the underlying principle is the same: by understanding the structure of the problem, we can make precise and impactful decisions even when information is limited. The success of both compressed sensing and venture capital depends not on having all the data, but on knowing which data matters.
If you're a founder or an investor who has been thinking about this, I'd love to hear from you.
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