[ad_1]

…and what not!

For any graph, the line on the graph forms a slope. The slope is given by the equation `(y2 — y1) / (x2 — x1)`

, where y2 is the value on the horizontal line that corresponds to x2 and y1 corresponds to x1.

For a set of graphs, you can find which graph has the steepest slope. For the graphs above, the steepest is the graph with a slope of 2. The highest value is the steepest. Slopes that are almost standing are steeper then slopes that are almost flat.

*Don’t keep a crawling baby on a very steep slope. A very gentle slope is okay.*

Given a graph, it’ll be nice to know the slope at any interval. Just get any 2 points of your choice on the x axis `[x2,x1]`

, find the corresponding values on the y axis `[y2,y1]`

, and find the `slope`

.

Lipschitz simply says that for a given graph, there’s always the steepest point, L.

Since the steepest slope , L, in a graph has the highest value, then all other slopes at any point in the graph are not as steep as L (or there might be other slopes that are just as steep as L, but not steeper than L).

If a graph is 2 lipschitz, it means that the steepest slope on that graph has a value of 2 (which means that other slopes in the graph are lesser than 2 or equal to 2). If it’s 43 lipschitz, it means that the steepest slope on that graph has a value of 43 (which means that other slopes in the graph are lesser than 43, and some might be equal to 43, but none is more than 43).

For a graph, `slope = (y2 — y1) / (x2 — x1)`

. But there’s the steepest slope , L, and all other slopes are lesser than L or some might be equal to L

ie `slope ≤ L`

which means `(y2 — y1) / (x2 — x1) ≤ L`

. But we’re not interested in negative slopes or positive slopes, just the values , so we’ll take the absolute values (ie remove any negative signs)

`|y2 — y1"https://medium.com/"x2 — x1| ≤ L`

multiplying both sides by `|x2 — x1|`

,

`|y2 — y1| ≤ L * |x2 — x1|`

, which gives us the lipschitz equation.

For very steep slopes, you’ll find out that there’s a very tiny range of x values (ie 0 to 1)and a large range of y values (0 to 100000000) . Therefore a small increase in x would cause a large increase in y, which might push your prediction towards the wrong classification.

On the other hand, for very gentle slopes, a very tiny change to x causes an even tinier change in y, which might be useful if the tiny changes are noises on your input data.

[ad_2]

Source link