Homeworks for the course CS 5354/CS 4365, Fall 2022

Submit solutions via email to vladik@utep.edu.

1. **(Due September 1)** We say that a function f(x) is
*scale-shift-invariant* if for every λ > 0, there
exists a value y_{0} for which y = f(x) implies that y' =
f(x'), where y' = y + y_{0} and x = λ * x.

Write down, in detail, the proof that if a differentiable function is scale-shift-invariant, then f(x) = A + a * ln(x) for some A and a.

*Comment.* Follow the pattern of proof about scale-scale
invariance (in the handout) and shift-scale invariance (what we did
in class). The main ideas of this proof are given in the Appendix
to paper 6, you just need to describe it in all the detail.

2. **(Due September 8)** Suppose that you know the values of
some quantity v in three points x_{1}, x_{2}, and
x_{3}, these values are v_{1} = 1, v_{2} =
2, and v_{3} = 3. Based on this information, you want to
use the inverse distance weighting technique (as described in Paper 3
and as we described in class) to predict the value v of this
quantity as a point x for which d(x_{1}, x) = 10,
d(x_{2}, x) = 20, and d(x_{3}, x) = 30. Take a =
−1.

3. **(Due September 8)** Use calculus to find the value x for
which the following function attains is minimum x^{2} + 2 *
x + 1. What is the value of this minimum?

4. **(Due September 13)** Describe a function y = 1/(1 +
exp(−x)) as a composition of invariant functions.
*Comment.* This function is actively used in neural
networks.

5. **(Due September 13)** Describe a function
√(x^{2} + x^{4}) as a scale-invariant
combination of two scale-scale-invariant functions. *Hint:*
use the fact that x^{4} = (x^{2})^{2}.

6. **(Due September 13)** Prove that the family of all cubic
polynomials

7. **(Due September 13 for extra credit, due September 20 for
regular credit)** As alternatives, let us consider families of
the type {C*f(x)}_{C}, where f(x) is fixed and C can take
any value. Let us define scaling T_{λ} as an
operation that transforms a family {C*f(x)}_{C} into a new
family {C*f(λ*x)}_{C}. Prove that if an optimality
criterion on the set of all such alternatives is final and
scale-invariant, then each function which from the optimal family
is a power law f(x) = A
* x^{a}. *Hint:* first, follow the general proof that
we had in class about the function optimal with respect to a
T-invariant criterion, and then use the result that we proved in
class -- that every scale-scale-invariant function is described by
the power law.

*Second hint:* Follow the logic of a
similar proof that we had in class.

8. **(Due September 15)** Submit a report on the progress of
your project.

9. **(Due September 29)** Use what we learned so far to explain
the following two empirical formulas: y = x^{2} *
e^{−kx} and y = x^{0.3} * ln(x).

10. **(Due September 29)** Show that the class of all functions
obtained from 1/(1 + k * exp(−x)) by fractional-linear
transformations is shift-invariant.

11. **(Due September 29)** Show that for each a, the class of
all functions obtained from 1/(1 + x^{a}) by
fractional-linear transformations is scale-invariant.

12. **(Due November 10)** Prepare handwritten "cheat sheet" for
Test 2, with no more than 5 double-size pages, scan it, and send to
the instructor. Students who got A on Test 2 do not need to do
it.