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Posted on July 7, 2014

neural networks , deep learning , representations , NLP , recursive neural networks

Introduction

In the last few years, deep neural networks have dominated pattern recognition. They blew the previous state of the art out of the water for many computer vision tasks. Voice recognition is also moving that way.

But despite the results, we have to wonder… why do they work so well?

This post reviews some extremely remarkable results in applying deep neural networks to natural language processing (NLP). In doing so, I hope to make accessible one promising answer as to why deep neural networks work. I think it’s a very elegant perspective.

A neural network with a hidden layer has universality: given enough hidden units, it can approximate any function. This is a frequently quoted – and even more frequently, misunderstood and applied – theorem.

It’s true, essentially, because the hidden layer can be used as a lookup table.

For simplicity, let’s consider a perceptron network. A perceptron is a very simple neuron that fires if it exceeds a certain threshold and doesn’t fire if it doesn’t reach that threshold. A perceptron network gets binary (0 and 1) inputs and gives binary outputs.

Note that there are only a finite number of possible inputs. For each possible input, we can construct a neuron in the hidden layer that fires for that input, 1 and only on that specific input. Then we can use the connections between that neuron and the output neurons to control the output in that specific case. Cheap Sale Cost Sister Cat Print Mini Skirt Taupe/chestnut Paul amp; Joe The Cheapest For Sale Best Store To Get Online Discount Fashion Style NlE5doqA

And so, it’s true that one hidden layer neural networks are universal. But there isn’t anything particularly impressive or exciting about that. Saying that your model can do the same thing as a lookup table isn’t a very strong argument for it. It just means it isn’t impossible for your model to do the task.

Universality means that a network can fit to any training data you give it. It doesn’t mean that it will interpolate to new data points in a reasonable way.

No, universality isn’t an explanation for why neural networks work so well. The real reason seems to be something much more subtle… And, to understand it, we’ll first need to understand some concrete results.

I’d like to start by tracing a particularly interesting strand of deep learning research: word embeddings. In my personal opinion, word embeddings are one of the most exciting area of research in deep learning at the moment, although they were originally introduced by Bengio, et al. more than a decade ago. 3 Beyond that, I think they are one of the best places to gain intuition about why deep learning is so effective.

The Artful Amoeba
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If you stumbled one midsummer on the melting snow in the image below, what would you imagine produced the strange color?

Translated German caption: "Snow area with Chlamydomonas nivalis (snow blood) near Abisko (Northern Sweden)" Creative Commons Ökologix. Click image for license and source.

Here's another example with a pinker hue, from further out.

Creative Commons Will Beback. Click image for license and source.

Here's a poorer example that I stumbled on myself on July 4, 2011 on Long's Peak in Colorado:

When an 1818 British expedition led by Captain John Ross tasked with finding the Northwest Passage stumbled onto "extensive patches" of this stuff near Greenland's Cape York in Baffin Bay, the Times of London confidently declared it to be iron-nickel meteorite detritus. In reality, the Scottish botanist Womens Selected Premium Bademantel 95 cm Bathrobe Schiesser Purchase Online Really For Sale GruZtBN9Wb
-- he of Brownian motion fame -- suggested in an appendix to Ross's mission report that same year the color could be due to an alga, a photosynthetic microbe. And it was.

If there's one thing Earth has taught us, it's that if a surface or substrate is ever wet, something will grow. And, despite near-zero temperatures, acidity, solar irradiation, and what must be frankly admitted to be minimal nutritional value, snow is no exception. Over 60 species of algae alone dwell there , and no doubt more await discovery. Scientists just announced this May the discovery of a new species from Colorado snow that they suggest could be a source of biofuel feedstock for northern climates where other algae cannot thrive.

By far, the most common species of snow alga is Chlamydomonas nivalis , which colors snow red or pink. With their pair of front-mounted flagella, they ply the films of water found in melting snow drifts. Midsummer is the best time of the year to see them, if you live in a high-altitude or Arctic clime with snowbanks that stubbornly refuse to yield to the sun.

Yet surprsingly, active C. nivalis cells are not pink when you look at them under the microscope. Here's what a different species of Chlamydomonas looks like swarming in water to give you the idea.

Here's a closeup alongside a slinky green alga called Euglena . The homely, roundish cells are Chlamydomonas , and you can see both their paired flagella and the cells' various organelles (aren't you glad our bodies aren't transparent?):

“Concept-Formation,” Womens Escape Swimsuit Miraclesuit Excellent Cheap Online Factory Outlet s75grq
, 11–12

Observe the multiple role of measurements in the process of concept-formation, in both of its two essential parts: differentiation and integration. Concepts cannot be formed at random. All concepts are formed by first differentiating two or more existents from other existents. All conceptual differentiations are made in terms of (i.e., characteristics possessing a common unit of measurement). No concept could be formed, for instance, by attempting to distinguish long objects from green objects. Incommensurable characteristics cannot be integrated into one unit.

Tables, for instance, are first differentiated from chairs, beds and other objects by means of the characteristic of , which is an attribute possessed by all the objects involved. Then, their particular kind of shape is set as the distinguishing characteristic of tables—i.e., a certain category of geometrical measurements of shape is specified. Then, within that category, the particular measurements of individual table-shapes are omitted.

Please note the fact that a given shape represents a certain category or set of geometrical measurements. Shape is an attribute; differences of shape—whether cubes, spheres, cones or any complex combinations—are a matter of differing measurements; any shape can be reduced to or expressed by a set of figures in terms of . When, in the process of concept-formation, man observes that shape is a commensurable characteristic of certain objects, he does not have to measure all the shapes involved ; he merely has to observe the element of .

Similarity is grasped ; in observing it, man is not and does not have to be aware of the fact that it involves a matter of measurement. It is the task of philosophy and of science to identify that fact.

“Concept-Formation,” , 13–14

A commensurable characteristic (such as shape in the case of tables, or hue in the case of colors) is an essential element in the process of concept-formation. I shall designate it as the “Conceptual Common Denominator” and define it as “The characteristic(s) reducible to a unit of measurement, by means of which man differentiates two or more existents from other existents possessing it.”

The distinguishing characteristic(s) of a concept represents a specified category of measurements within the “Conceptual Common Denominator” involved.

New concepts can be formed by integrating earlier-formed concepts into wider categories, or by subdividing them into narrower categories (a process which we shall discuss later). But all concepts are ultimately reducible to their base in perceptual entities, which are the base (the given) of man’s cognitive development.

“Concept-Formation,” , 15

When concepts are integrated into a wider one, the new concept includes the characteristics of its constituent units; but their distinguishing characteristics are regarded as omitted measurements, and one of their common characteristics determines the distinguishing characteristic of the new concept: the one representing their “Conceptual Common Denominator” with the existents from which they are being differentiated.

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