In this way, a line is made to fill a plane. This process is repeated infinitely many times, resulting in a curve that passes through every point of the original plane square. At this stage, the original pattern, repeated 9 times, is connected together by 8 links, forming a single curve. Then each sub square is divided again into 9 sub squares whose centers are all connected by lines. Lines connect the centers of each of the sub squares.
The construction of Peano’s curve proceeds by taking a square and dividing it into 9 equal sub squares. When the peripatetic mathematician Guiseppe Peano learned of Cantor’s result for the mapping of 2D to 1D, he sought to demonstrate the correspondence geometrically, and he constructed a continuous curve that filled space, publishing the method in Sur une courbe, qui remplit toute une aire plane in 1890. There is infinite room within the reals to accommodate the countably infinite number of repeating-decimal numbers. Numbers that have two equivalent representations can be shifted to the right to replace other numbers that are shifted further to the right, and so on to infinity. Originally, Cantor was interested in the behavior of functions at these points, but his interest soon shifted to the properties of the points themselves, which became his life’s work as he developed set theory and transfinite mathematics.įig.
#Unbound reality fragment series
He had found that even though the series might converge to a function almost everywhere, there surprisingly could still be an infinite number of points where the convergence failed. Cantor published a paper early in 1872 on the question of whether the representation of an arbitrary function by a Fourier series is unique. He received his doctorate in 1867 and his Habilitation in 1869, moving into a faculty position at the University of Halle and remaining there for the rest of his career. In 1863, he enrolled at the University of Berlin where he sat on lectures by Weierstrass and Kronecker. Georg Cantor (1845 – 1918) was born in Russia, and the family moved to Germany while Cantor was still young. In fact, we owe our evolutionary existence to this effect! Therefore, deep dimensionality reduction is a way to bring complex data down to a dimensionality where our intuition can be applied to “explain” the data.īut what is a dimension? And can you find the “right” dimensionality when performing dimensionality reduction? Once again, our intuition struggles with these questions, as first discovered by a nineteenth-century German mathematician whose mind-expanding explorations of the essence of different types of infinity shattered the very concept of dimension. Even the topology of landscapes in high dimensions is unintuitive-there are far more mountain ridges than mountain peaks-with profound consequences for dynamical processes such as random walks (see my Blog on a Random Walk in 10 Dimensions). For instance, in very high dimensions almost all random vectors in a hyperspace are orthogonal, and almost all random unit vectors in the hyperspace are equidistant.
Many of the things we take for granted in our pitifully low dimension of 3 (or 4 if you include time) just don’t hold in high dimensions. Second, the geometry of high dimension is highly unintuitive. Deep learning dimensionality reduction seeks to find the dependences-many of them nonlinear and non-single-valued (non-invertible)-and to reject the noise channels. While many others may be pure noise-or at least not relevant to the pattern. And many, or even most, of those entries may not be independent. Trying to visualize such high dimensions may sound mind expanding, but it is really just saying that a data problem may have hundreds or thousands of different data entries for a single event. There are two driving reasons to reduce the dimensionality of data:įirst, typical dimensionalities faced by machine learning problems can be in the hundreds or thousands. Hence, one of the principle functions of machine learning is to reduce the dimensionality of the data to lower dimensions-a process known as dimensionality reduction. Machine learning is characterized, more than by any other aspect, by the high dimensionality of the data spaces it seeks to find patterns in.
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