What is curse of dimensionality? How does it affect distance and similarity measures?

What is curse of dimensionality? How does it affect distance and similarity measures?



-Refers to various phenomena that arise when analyzing and organizing data in high dimensional spaces
-Common theme: when number of dimensions increases, the volume of the space increases so fast that the available data becomes sparse
-Issue with any method that requires statistical significance: the amount of data needed to support the result grows exponentially with the dimensionality
-Issue when algorithms don't scale well on high dimensions typically when O(nkn)O(nkn)
-Everything becomes far and difficult to organize

Illustrative example: compare the proportion of an inscribed hypersphere with radius rr and dimension d to that of a hypercube with edges of length 2r2r
- Volume of such a sphere is Vsphere=2rdπd/2dΓ(d/2)Vsphere=2rdπd/2dΓ(d/2)
- The volume of the cube is: Vcube=2rdVcube=2rd
As d increases (space dimension), the volume of hypersphere becomes insignificant relative to the volume of the hypercube:

limd→∞VsphereVcube=πd/2d2d−1Γ(d/2)=0
limd→∞VsphereVcube=πd/2d2d−1Γ(d/2)=0

- Nearly all of the dimensional space is far away from the center
- It consists almost entirely of the corners of the hypercube, with no middle!

Popular posts from this blog

After analyzing the model, your manager has informed that your regression model is suffering from multicollinearity. How would you check if he's true? Without losing any information, can you still build a better model?

After analyzing the model, your manager has informed that your regression model is suffering from multicollinearity. How would you check if he's true? Without losing any information, can you still build a better model? Answer: To check multicollinearity, we can create a correlation matrix to identify & remove variables having correlation above 75% (deciding a threshold is subjective). In addition, we can use calculate VIF (variance inflation factor) to check the presence of multicollinearity. VIF value <= 4 suggests no multicollinearity whereas a value of >= 10 implies serious multicollinearity. Also, we can use tolerance as an indicator of multicollinearity. But, removing correlated variables might lead to loss of information. In order to retain those variables, we can use penalized regression models like ridge or lasso regression. Also, we can add some random noise in correlated variable so that the variables become different from each other. But, adding noise m...
Read more

Is rotation necessary in PCA? If yes, Why? What will happen if you don't rotate the components?

Is rotation necessary in PCA? If yes, Why? What will happen if you don't rotate the components? Answer: Yes, rotation (orthogonal) is necessary because it maximizes the difference between variance captured by the component. This makes the components easier to interpret. Not to forget, that's the motive of doing PCA where, we aim to select fewer components (than features) which can explain the maximum variance in the data set. By doing rotation, the relative location of the components doesn't change, it only changes the actual coordinates of the points. If we don't rotate the components, the effect of PCA will diminish and we'll have to select more number of components to explain variance in the data set.
Read more

What does Latency mean?

What does Latency mean? Latency is a networking term to describe the total time it takes a data packet to travel from one node to another. In other contexts, when a data packet is transmitted and returned back to its source, the total time for the round trip is known as latency. Latency refers to time interval or delay when a system component is waiting for another system component to do something. This duration of time is called latency.
Read more