I was just wondering what a negative inner product denotes when we are talking about functions being lined up with each other? As in if <<f,g>> = -5 and <<h,g>> = 2, can we imply anything by comparing their magnitudes and signs?
keenan
Sure—for instance, $\langle\langle f, -f, \rangle\rangle$ will be negative (unless $||f||=0$), but "large" in value (relative to the norm of $f$, say). So that sort of indicates that the two functions are "well lined up, but pointing in opposite directinos" (intuitively). Of course, if we are considering functions that are nonnegative everywhere (as is usually the case with, say, image intensity), then there is no way to get a negative $L^2$ inner product. In this case, the inner product really does just tell us how well the two functions line up (along with something about their overall magnitudes...).
cche
Is computing the inner product here similar to computing the cross-correlation between two functions? For example, if we want to find how similar two images?functions? are, we can calculate the correlation. Based on the equations above, if we slide one of the functions through another, and calculate the integral, we can obtain the correlation. Am I correct or I misunderstood the function here?
keenan
@cche: Yes, in fact the cross-correlation is sometimes called the "sliding dot product!" When restricted to just a single parameter value (no shift), this becomes the usual $L^2$ inner product.
Melody
I was wondering what does the inner product of two functions represent? According to the example and demonstration, the smaller value of the inner product means two functions not lining up. In other word, can we consider that two functions with similar trend will lead to large value, on the other side, the inner product of two functions with opposite direction will be smaller?
keenan
@Melody Yes, that's exactly the right intuition! Though it's important to be careful about the word "smaller." For instance, -100 is "smaller" than 5 in the sense that -100 < 5. But it's not smaller in magnitude, since $|-100| > |5|$. Likewise, you may have two functions that "look" extremely similar, but are "pointing" in different directions. For instance, $5 f(x)$ and $-100 f(x)$ will have an inner product that is large in magnitude, even though it's very negative in value.
I was just wondering what a negative inner product denotes when we are talking about functions being lined up with each other? As in if <<f,g>> = -5 and <<h,g>> = 2, can we imply anything by comparing their magnitudes and signs?
Sure—for instance, $\langle\langle f, -f, \rangle\rangle$ will be negative (unless $||f||=0$), but "large" in value (relative to the norm of $f$, say). So that sort of indicates that the two functions are "well lined up, but pointing in opposite directinos" (intuitively). Of course, if we are considering functions that are nonnegative everywhere (as is usually the case with, say, image intensity), then there is no way to get a negative $L^2$ inner product. In this case, the inner product really does just tell us how well the two functions line up (along with something about their overall magnitudes...).
Is computing the inner product here similar to computing the cross-correlation between two functions? For example, if we want to find how similar two images?functions? are, we can calculate the correlation. Based on the equations above, if we slide one of the functions through another, and calculate the integral, we can obtain the correlation. Am I correct or I misunderstood the function here?
@cche: Yes, in fact the cross-correlation is sometimes called the "sliding dot product!" When restricted to just a single parameter value (no shift), this becomes the usual $L^2$ inner product.
I was wondering what does the inner product of two functions represent? According to the example and demonstration, the smaller value of the inner product means two functions not lining up. In other word, can we consider that two functions with similar trend will lead to large value, on the other side, the inner product of two functions with opposite direction will be smaller?
@Melody Yes, that's exactly the right intuition! Though it's important to be careful about the word "smaller." For instance, -100 is "smaller" than 5 in the sense that -100 < 5. But it's not smaller in magnitude, since $|-100| > |5|$. Likewise, you may have two functions that "look" extremely similar, but are "pointing" in different directions. For instance, $5 f(x)$ and $-100 f(x)$ will have an inner product that is large in magnitude, even though it's very negative in value.