In the beginning of the training process of a Neural Network, it's parameters, for example the weights in a Fully Connected Layer, have to initialized. There is a wide variety of schemes, how you can do that.   One of the simplest one is a constant initialization: All weights are set to the same value.  The output of each neuron in the forward pass is then of course the same. But how bad is that? Given that there are other layers and regularization is used, shouldn't that mitigate this process and still allow the network to learn properly?  In this context I also often hear the phrase: "We don't do constant initialization to break symmetry" - but what that does actually mean, why do we want that?
Why doesn't backpropagation work when you initialize all the weight the same value (say 0.5), but works fine when given random numbers?   Shouldn't the algorithm calculate the error and work from there, despite the fact that the weights are initially the same?
How do I bind a WPF DataGrid to a variable number of columns? My WPF application generates sets of data which may have a different number of columns each time.  Included in the output is a description of each column that will be used to apply formatting.  A simplified version of the output might be something like:  class Data {     IList&lt;ColumnDescription&gt; ColumnDescriptions { get; set; }     string[][] Rows { get; set; } }   This class is set as the DataContext on a WPF DataGrid but I actually create the columns programmatically:  for (int i = 0; i &lt; data.ColumnDescriptions.Count; i++) {     dataGrid.Columns.Add(new DataGridTextColumn     {         Header = data.ColumnDescriptions[i].Name,         Binding = new Binding(string.Format("[{0}]", i))     }); }   Is there any way to replace this code with data bindings in the XAML file instead? Can I be admitted into a PhD program with non-thesis MS degree? If I do a non-thesis MS program in statistics, will I be eligible for an admission in to a PhD program in Statistics later in my life? The MS program I am looking into is a non-thesis program but I still have to complete a research project requirement (which involves oral presentation as well as some form of a written report) towards the end of the program...I am still not sure if it can lead to a PhD program though.  Thanks! When did The Doctor start running so much? Thanks to Netflix Streaming, I've been watching some random classic Doctor Who stories.  One thing that strikes me as different between the modern Who and old Who was that in modern Who, they do an awful lot of running.  It's become a sort of running gag that they've referenced in-universe.  In the classic Who I've seen, he tends to drive around quite a bit instead.  In addition, The Doctor in modern Who is incredibly reliant on his sonic screwdriver, which I often don't see at all in classic Who stories.  Did the Doctor running and the sonic screwdriver get introduced over time in classic Who, or were they both added as part of the modern Who revival? I'm an on and off user of LaTeX so I'm not familiar with every package out there. Having said that, whenever I use LaTeX and need to accomplish a certain goal (i.e. ), I search around and find some kind of solution that involves a package that I didn't know about. I think that's great, I learn something new, and it goes in my template file.  Every now and then, though, my search comes up with a package that, after much wrangling, I find out that it's obsolete. An example of this is just recently I came across the subfigure package and wanted to try it out. Soon after I found out from here that it has become . I usually keep my TeX up-to-date because I like keeping up with new XeLaTeX features.   My question is, is there a map or timeline of what TeX packages should be avoided because there are better ways of doing things? An idea I have is some kind of list of all packages split by what they accomplish, then cancelling out packages whose features are incorporated into and improved upon by newer packages.  Edit: another example is lscape versus pdflscape. The latter implements rotating pages in the final PDF not just for pdftex but also for other drivers such as xetex. The only way I found out about this was Googling latex lscape rotate xelatex, for which the second link leads to the  (PDF).  Great answers so far. I'll check them out and see which one works best for me.  Edit 2: Turns out that my question is a bit of a dupe in the sense that a great answer came up on an unrelated question. The nag package . Gotta try it! I want to implement an addition key layer, so that I can access non-alphanumeric keys without having to move my wrists.  I've been using Autokey for this purpose, but the experience was unsatisfactory: it had occasional lags and let original keystrokes slip into certain apps.  So I need a low-level solution. The site description in the hero header is wrong for Meta sites The hero header contains the same description for the Main and Meta sites. Here's the one for MSO:           Meta Stack Overflow is a question and answer site for professional and enthusiast programmers. Join them; it only takes a minute:    This can easily confuse people who are new to the network, so I feel this should be changed to something Meta specific. Perhaps re-use the text from the sidebar, that has a good description:     This site is intended for bugs, features, and discussion of Stack Overflow and the software that powers it. You must have an account on Stack Overflow to participate. Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$.   I've tried starting with just $a \leq c$ and using field properties to reconstruct the inequality, however I haven't been able to make it work. I also tried making the negatives positive and stripping the inequalities and making something happen but again I don't know if that's a proper rule and it didn't seem to get me anywhere. What is a good way to transform Cyclic Ordinal attributes? Look at this if block:  #!/bin/bash  str="m.m" if [[ "${str}" =~ "m\.m" ]]; then     echo "matched" else     echo "not matched"     exit 1 fi  exit 0   This should print "matched", but it doesn't. Where am I going wrong? Is there any way to force SQL Server NOT to mark subscriptions as inactive, ever?  It happens sporadically when there are connection issues and I don't want to have to reinitialize the subscription every time.  Note, I'm not talking about the subscriptions being marked as expired...just as inactive.  We have retention set to 0 (infinite). I'd rather have a big distribution database than have to reinitialize a 15GB database to a subscriber over the WAN. Anyway, the real issue here is that they are getting marked as inactive within just a few hours sometimes...or sometimes a few days...it seems sporadic. Is there any way to tell WHY SQL decided in its wisdom to mark them inactive? What should I do if a user posts sensitive information as part of a question or answer? I've observed several incidents where users inadvertently posted detailed, sensitive information. For example, a Server Fault user who posted a question with working login credentials to his actual production server, or the Stack Overflow user who posted actual medical records as example data for his question.   What should be the procedure for handling such cases? How to disable the "highlight changes" feature? Our admins deployed a new version of Notepad++ (version 6.3) on our machines, and this version is highlighting all changes done in a file by default. It does so by setting the background color of every modified line to a bright orange, which I find very distracting.  We  have dozens of plugins installed, but I double checked and we do not have the "Change Markers" plugin (which could be the culprit).  Any idea where I can disable this feature? Changing the highlight color could also be an option. When I'm not exploring fresh caves or lands in minecraft, I'm busy trying to find deposits of diamond, gold and iron deep in the earth. So far I've just been digging around just above bedrock on a whim to look for these rare minerals but I've started thinking that I should be taking a more controlled, efficient approach to mining.  What is the optimal mining strategy that yields the most rare minerals while minimizing blocks removed and time spent digging and traveling?  Edit: The existing answers are great for mining, but no one has addressed travel time to and from the mine which is usually ~60 blocks downward and incredibly tedious. Solutions? What's the intuition behind difference of $\neg(S\land E)$ and $\neg S \land \neg E$? Let $S$ represent the English statement "Sales will go up."And let $E$ represent the English statement "Expenses will go up." Now my problem is that I have trouble telling the difference between these two statements: $\neg(S\land Q)$ and $\neg S \land \neg E$. When these two statements are translated into English they sound,to me, the same. However, when you look at their truth tables they are different.   Here is my best translation of these two statements:  1) $\neg(S\land E)$ represented in English, "Both sales and expenses will not go up."(Correct me if I translated it wrong)  2)$\neg S \land \neg E$ represented in English, "Expenses will not go up and sales will not go up."(Again, correct me if I'm wrong)  I still do not understand the difference between these two statements in the English representation. They both sound the same thing to me.  Once more, I have the same trouble I have with understanding with the or-connective( $\lor$ ). Again let E and S be the same statements as before.Tell me if I might have translated them wrong.  1) $\neg (S \lor E)$ represented in English is "Neither sales nor expenses will go up."  2)$\neg S \lor \neg E$ represented in English is "Sales will not go up, or Expenses will not go up."  Yet I still have trouble understanding the difference between the English translation.  In short, can you give me some alternative example that better explains the difference, something intuitive that gives me a clear picture of the difference of the meaning between the statements I showed with the connective-or and connective-and. Thank you. Objects disappear when looking through camera Suppose that $X_1,X_2,\ldots,X_n$ is a random sample from a distribution with probability function:  $$p_X(x)=\begin{cases} 1/2 &amp;x=-1,1 \\ 0 &amp; \text{otherwise} \end{cases}$$  Now if we define the average   $\bar{X}= \frac{1}{n} \sum X_i$, I need to show that that for an odd $n$ the probability function for $\bar{X}$ is:  $$p_{\bar{X}} (x)= \frac{\binom{n}{\frac{n}{2} (x+1)}}{2^n}$$ for $x=\pm 1/n,\pm 3/n,\ldots,\pm 1$, $0$ otherwise.    Since $P[\sum X_i=k] =P [\bar{X}=k/n ]$, I thought it would be easier to first derive the probability function for the sum. Since they are still $2^n$ n-tuples and each one is equilikely, I need to count the ones whose sum is $k$ for:  $k=\pm 1,\pm 3,\ldots \pm n-2,\pm n$ (since even numbers are not a feasible combination)  The problem here though is that I do not immediately recognize how I could count all these tuples. Could you please help me with this? If it is easier to proceed another way, I am of course all ears. Adjust space between entries in LoF and LoT Equals(=) vs. LIKE When using SQL, are there any benefits of using = in a WHERE clause instead of LIKE?  Without any special operators, LIKE and = are the same, right? Test if a string contains any of the strings from an array Prove that the difference between two rational numbers is rational This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt:     Let $a$ and $b$ be rational numbers. Therefore, \begin{align}  a=\frac{\lambda}{\beta},\:b=\frac{\xi}{\zeta},\:\ni\:\lambda,\beta,\xi,\zeta\in\mathbb{Z},\tag{1} \end{align} which gives us \begin{align}  a-b=\frac{\lambda}{\beta}-\frac{\xi}{\zeta}=\frac{\lambda\zeta-\xi\beta}{\beta\xi}.\tag{2} \end{align}   So I have shown that $a$ and $b$ are rational numbers which, by definition, can be represented by the quotient of two integers. But now how do I tackle the problem of the difference? By definition the difference between two integers is an integer. Does that require that this difference is thus rational?  Thank you for your time,