Gradient Ascent #6

Invariance, perception, accidental viewpoints

Welcome to the 6th edition of Gradient Ascent. I’m Albert Azout, a prior entrepreneur and current Partner at Cota Capital. On a regular basis I encounter interesting scientific research, startups tackling important and difficult problems, and technologies that wow me. I am curious and passionate about machine learning, advanced computing, distributed systems, and dev/data/ml-ops. In this newsletter, I aim to share what I see, what it means, and why it’s important. I hope you enjoy my ramblings!

Is there a founder I should meet?

Send me a note at

Want to connect?

Find me on LinkedIn, Angelist, Twitter

In this edition, some overly philosophically thoughts about chairs…

While quirky and laughable, the Beuchet chair (below) tells an interesting story about human intelligence, and maybe humanity. In perceiving an object, our brains make a (presumably) valid assumption: the object we perceive and the information delivered to us from our perceptual process (light reaching our eyes, processed by our brains, etc) are independent (i.e. there is no dependency between the integrity of the object and our point of view, the object is always the same regardless of where we are standing).

The chair above invalidates this assumption, in the case that we happen upon an accidental viewpoint (above, right), and perceive the three-dimensional structure of a non-existent chair.

The above independence assumption (object and viewpoint) is useful because in practice, it holds most of the time, and our brain thus relies on objects being independent of our vantage point and the illumination (source).

This is the generic viewpoint assumption. The observer is never expected in a special position relative to the scene, but rather all positions would hold the object’s structural properties in tact. This assumption represents an invariance that exists deep within our visual system. An innate, embedded construct of our perception. We assume that as we move around the object (virtually, in our minds), our vantage point changes, but the overall generative process (e.g. object position and structure) is unchanged, unalterable.

Many interesting perceptual oddities extend from accidental viewpoints, including ambiguous images and multistable perception. The Penrose triangle (below), viewed from an accidental viewpoint, generates an impossible object (caused by our natural desire to interpret 2D drawings as three-dimensional objects).

These ideas (from Penrose) were exploited in M. C. Escher’s incredible work (see Relativity below).

In AI and computer vision, the fundamental object recognition problem is to classify objects, independent of their viewpoint (e.g. to find viewpoint invariant representations). Convolutional neural nets (CNNs) accomplish this via translational invariance (weight sharing) and lots of training data.

Learning viewpoint invariant representations involves compressing objects into abstract (and disentangled) features (Deep Learning and the Informational Bottleneck Principle). CNN’s do this via layers of neurons, which extract higher-order features from pixel level data. The perceptual process (and compression) involves removing the redundant and the superfluous, keeping only that which is useful for recognition. In Beuchet’s chair, the presumably redundant information is our mind’s own viewpoint. What if our our brains needed to consider every possible viewpoint and illumination condition in order to recognizing an object? Way too costly. A chair must be a chair, from wherever we stand.

This assumption holds together until we encounter an accidental, eccentric viewpoint—the rare perspective in which we see a cohesive object that is not really there. I think the inverse of this scenario perhaps has caused humanity the most pain, whereby we choose to remain fixated on an incorrect viewpoint, a psychological one, held in the mind’s eye, no longer accidental but nonetheless a persistent illusion.

Cheers to 🍷 always seeing things from multiple angles 📐.

Non Sequiturs

Really amazing and well-written book by Vaclav Smil


While the author of this publication is a Partner with Cota Capital Management, LLC (“Cota Capital”), the views expressed are those of the writer author alone, and do not necessarily reflect the views of Cota Capital or any of its affiliates. Certain information presented herein has been provided by, or obtained from, third party sources. The author strives to be accurate, but neither the author nor Cota Capital do not guarantees the accuracy or completeness of any information.

You should not construe any of the information in this publication as investment advice. Cota Capital and the author are not acting as investment advisers or otherwise making any recommendation to invest in any security. Under no circumstances should this publication be construed as an offer soliciting the purchase or sale of any security or interest in any pooled investment vehicle managed by Cota Capital. This publication is not directed to any investors or potential investors, and does not constitute an offer to sell — or a solicitation of an offer to buy — any securities, and may not be used or relied upon in evaluating the merits of any investment.

The publication may include forward-looking information or predictions about future events, such as technological trends. Such statements are not guarantees of future results and are subject to certain risks, uncertainties and assumptions that are difficult to predict. The information herein will become stale over time. Cota Capital and the author are not obligated to revise or update any statements herein for any reason or to notify you of any such change, revision or update.