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Estimating Unknown Sparsity in Compressed Sensing is a paper about sparse signal. I am just learning the concepts. In the first paragraph, it says that when the number of observation data samples $n$ is less than the signal dimension $p$, then the desired signal $x$ is sparse. I have never come across he term signal dimension. Can somebody please explain what is signal dimension ?

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2 Answers 2

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The signal dimension $p$ refers to the dimension of the signal. The paper in question specifies that it is working in $R^p$, not ($R^1$) because the author wants to take into account multiple dimensions. For example, if your data is a geographic location, you have dimension $p=3$ because the complete description requires 3 numbers: longitude, latitude and altitude.

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  • $\begingroup$ That means it is the number of variables?Can you please explain with an example equation?Or is it like a vector auto regressive model? $\endgroup$
    – Ria George
    Commented Aug 6, 2014 at 2:05
  • $\begingroup$ No that does not mean the number of variables. I gave you an example of geographic location. If that is your variable, you need 3 numbers to completely describe the geographic location of an object: longitude, latitude and altitude. For example, point A is located at co-ordinates (x,y,z). It has nothing to do with vector auto-regressive model. It is not a model at all, just a description. $\endgroup$
    – rocinante
    Commented Aug 6, 2014 at 2:21
  • $\begingroup$ Are they the number of coefficients of a linear regression model, say auto regression model of order 2: AR(2) then is signal dimension p=2? $\endgroup$
    – Ria George
    Commented Aug 6, 2014 at 2:24
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    $\begingroup$ No. It is not the number of coefficients of any model. It is just the complete description of a single variable. It has nothing to do with the dimension of a model. AR(2) tells you nothing about the dimension of the signal. $\endgroup$
    – rocinante
    Commented Aug 6, 2014 at 2:36
  • $\begingroup$ Ok, so by looking at the signal is there a way to know what the dimension is? The thing is I am learning about sparse regression, and everywhere I get is that the signal dimension must be greater than number of samples observed. So, in this context how do I determine if the model is sparse or not if nothing is reflected in the functional form of the model equation?Shall be obliged for your insights into this matter. $\endgroup$
    – Ria George
    Commented Aug 6, 2014 at 2:41
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Signal dimension is just the dimension of the random variable that is the signal. "Dimension $p$" is a slightly sloppy way of stating the dimension of the support of the random variable is $p$. That is, it's a multivariate signal with $p$ components.

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  • $\begingroup$ Can you please explain with an example of what a multivariate signal is?How it looks like in eQuation form? $\endgroup$
    – Ria George
    Commented Aug 6, 2014 at 2:06
  • $\begingroup$ Color on a computer screen is a 3-dimensional signal of red, green, and blue. This would be represented as a vector $color = (red, green, blue) \in \mathbb{R}^3$. $\endgroup$ Commented Aug 6, 2014 at 2:36
  • $\begingroup$ Both of your answers are different. Does dimensionality means multivariate signal where each variable = dimension?So, in linear regression I will have 3 equations if dimension = 3?Can you please explain with AR model? $\endgroup$
    – Ria George
    Commented Aug 6, 2014 at 2:46
  • $\begingroup$ It's a matter of interpretation. You can think of "color" as one variable, or "red," "green," and "blue" as three separate variables. $\endgroup$ Commented Aug 6, 2014 at 2:51

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