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Multi-head attention lets a Transformer attend to different things at once. With input X (n tokens × d columns) and weight matrices of size d × d:
- Q = X × w_q, K = X × w_k, V = X × w_v
- Split the d columns into
num_headsheads of d_k = d / num_heads consecutive columns (head h uses columns h×d_k to (h+1)×d_k - 1). - For each head, run scaled dot-product attention on its columns: for token i, score_j = (Q_i · K_j) / √d_k over that head's columns, softmax over all n tokens j, and output = Σ weight_j × V_j (that head's columns).
- Put the heads' outputs back in their columns (concatenate) and multiply by w_o.
Implement multi_head_attention(x, w_q, w_k, w_v, w_o, num_heads).
Input
x: n × d input matrix.
w_q, w_k, w_v, w_o: d × d weight matrices.
num_heads: Number of heads (d is divisible by it).
Output
Return the n × d output matrix.
Constraints
1 ≤ n ≤ 30, 1 ≤ d ≤ 32, 1 ≤ num_heads ≤ d and d is divisible by num_heads
-5 ≤ each value ≤ 5
Examples
Example 1
Input:
x = [
[1.0, 0.0],
[0.0, 1.0]
]
w_k = [
[1.0, 0.0],
[0.0, 1.0]
]
w_o = [
[1.0, 0.0],
[0.0, 1.0]
]
w_q = [
[0.0, 0.0],
[0.0, 0.0]
]
w_v = [
[1.0, 0.0],
[0.0, 1.0]
]
num_heads = 1
Expected output:
[
[0.5, 0.5],
[0.5, 0.5]
]
Explanation
w_q is all zeros, so every score is 0 and each token averages the value rows: V = X, so both rows become [0.5, 0.5].
Example 2
Input:
x = [
[2.0, 4.0]
]
w_k = [
[1.0, 0.0],
[0.0, 1.0]
]
w_o = [
[0.0, 1.0],
[1.0, 0.0]
]
w_q = [
[1.0, 0.0],
[0.0, 1.0]
]
w_v = [
[1.0, 0.0],
[0.0, 1.0]
]
num_heads = 2
Expected output:
[
[4.0, 2.0]
]
Explanation
With a single token each head attends only to itself, so the concatenated output is V = [2, 4]. w_o swaps the two columns: [4, 2].
References
-
Example 1 Not runInput:
x = [ [1.0, 0.0], [0.0, 1.0] ] w_k = [ [1.0, 0.0], [0.0, 1.0] ] w_o = [ [1.0, 0.0], [0.0, 1.0] ] w_q = [ [0.0, 0.0], [0.0, 0.0] ] w_v = [ [1.0, 0.0], [0.0, 1.0] ] num_heads = 1Expected output:[ [0.5, 0.5], [0.5, 0.5] ]Your output -
Example 2 Not runInput:
x = [ [2.0, 4.0] ] w_k = [ [1.0, 0.0], [0.0, 1.0] ] w_o = [ [0.0, 1.0], [1.0, 0.0] ] w_q = [ [1.0, 0.0], [0.0, 1.0] ] w_v = [ [1.0, 0.0], [0.0, 1.0] ] num_heads = 2Expected output:[ [4.0, 2.0] ]Your output
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💡 C# Basics & Examples You Can Try Above
1. Declaring Variables and Constants
C# uses strongly-typed variable declarations. Use const for compile-time constants and readonly for runtime constants.
int age = 30;
double pi = 3.14159;
char grade = 'A';
string name = "Alice";
bool isActive = true;
// Constants
const int MaxUsers = 100;
const string Company = "CodeUtility";
2. Conditionals (if / switch)
Use if, else if, and switch for control flow.
int x = 2;
if (x == 1)
{
Console.WriteLine("One");
}
else if (x == 2)
{
Console.WriteLine("Two");
}
else
{
Console.WriteLine("Other");
}
switch (x)
{
case 1:
Console.WriteLine("One");
break;
case 2:
Console.WriteLine("Two");
break;
default:
Console.WriteLine("Other");
break;
}
3. Loops
C# supports for, while, and foreach loops.
for (int i = 0; i < 3; i++)
{
Console.WriteLine(i);
}
int n = 3;
while (n > 0)
{
Console.WriteLine(n);
n--;
}
4. Arrays
Arrays store fixed-size collections of elements of the same type.
int[] numbers = { 10, 20, 30 };
Console.WriteLine(numbers[1]);
5. List Manipulation
Use List<T> for dynamic collections.
List<int> nums = new List<int> { 1, 2, 3 };
nums.Add(4);
nums.Remove(2);
foreach (int n in nums)
{
Console.Write(n + " ");
}
6. Console Input/Output
Use Console.WriteLine and Console.ReadLine for basic I/O.
Console.Write("Enter your name: ");
string name = Console.ReadLine();
Console.WriteLine($"Hello, {name}!");
7. Functions
Define methods using a return type, name, and parameters.
int Add(int a, int b)
{
return a + b;
}
Console.WriteLine(Add(3, 4));
8. Dictionaries
Dictionary<TKey, TValue> stores key-value pairs.
Dictionary<string, int> ages = new Dictionary<string, int>();
ages["Alice"] = 30;
Console.WriteLine(ages["Alice"]);
9. Exception Handling
Handle runtime errors using try, catch, and finally.
try
{
throw new Exception("Something went wrong");
}
catch (Exception ex)
{
Console.WriteLine(ex.Message);
}
10. File I/O
Use File and StreamReader/StreamWriter for file operations.
File.WriteAllText("file.txt", "Hello File");
string text = File.ReadAllText("file.txt");
Console.WriteLine(text);
11. String Manipulation
C# strings support methods like Length, Substring, Contains.
string message = "Hello World";
Console.WriteLine(message.Length);
Console.WriteLine(message.Substring(0, 5));
Console.WriteLine(message.Contains("World"));
12. Classes & Objects
C# supports object-oriented programming through classes and objects.
class Person
{
public string Name;
public Person(string name) => Name = name;
public void Greet() => Console.WriteLine($"Hi, I'm {Name}");
}
Person p = new Person("Alice");
p.Greet();