![]() ![]() ![]() While it may seem innocuous (or maybe even conservative) to make a Gaussian assumption in high-dimensional Euclidean spaces, there can be unexpected practical consequences. to define the conditions under which the phenomena appear and claimed 4. Again, this isn’t true in two dimensions, but we’re imagining these points are in a high-dimensional space in this exercise.įerenc Huszár writes that we should think of high-dimensional Gaussians as “soap bubbles”, where most of the mass is on the outer shell of the bubble. page 156 of 3,: That it is an abstract, unintuitive mathematical. Daniel Kahneman coined the term misconceptions of chance to describe the phenomenon of people extrapolating large-scale patterns to samples of a much. When we sample from the Gaussians, most of the points lie on a thin ring around the mean. to describe and understand the new unintuitive phenomena that emerge in this regime in order to. In reality, however, the right panel is closer to reality. And if any of this stuff seems kind of unintuitive or daunting, or really on some level confusing- this wave-particle duality, this idea of a transfer of. Similar examples, with large datasets describing. There are many phenomena that follow this basic size-like idea including such phenomena as mass, length, and volume (and as we will see later, probability). What we might imagining is happening is plotted in the left panel below: three nice clouds of points, with most of the mass near the center of each cluster. Size is a number that we attribute to an object that obeys a specific, intuitive property: if we break the object apart, the sizes of the pieces should add up to the size of the whole. This example is adapted from that of Ferenc Huszár. Here, we’ll show plots in two dimensions for clarity, although clearly this phenomenon doesn’t hold in two dimensions. To consider how far astray our low-dimensional intuition can lead us in high-dimensional contexts, consider again sampling from a set of high-dimensional Gaussians (suppose $p=1,000$, for example). While there doesn’t seem to be a single agreed-upon definition for it, the term generally refers to the unintuitive behavior of high-dimensional spaces and the difficulty of accurately and efficiently modeling data in these spaces. So the color we see is the light that is reflected. It absorbs the rest (or at least the rest of the visible wavelengths). For example, a (yellow) banana is reflecting light of approximately 580 nm wavelength. : not readily learned or understood Working with a nonintuitive interface causes many programmers to make mistakes. This is one instance of a diffuse phenomenon that’s widely known as the “curse of dimensionality”. In a nutshell: The color we percieve an object to have, is the light that was reflected by that object. In words, this means that the squared norm of a Gaussian random vector will be proportional to the dimension of the Gaussian. Where we have used the fact that the sum of $p$ squared standard Gaussians is distributed as a $\chi^2$ random variable with $p$ degrees of freedom. ![]() The unintuitive nature of high-dimensional spaces ![]()
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