Multi-Head Self-Attention

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_heads heads 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_q = [
    [0.0, 0.0],
    [0.0, 0.0]
]
w_k = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_v = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_o = [
    [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_q = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_k = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_v = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_o = [
    [0.0, 1.0],
    [1.0, 0.0]
]
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

Run the sample tests to check your code without submitting.

Example 1 Not run

Input:

x = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_q = [
    [0.0, 0.0],
    [0.0, 0.0]
]
w_k = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_v = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_o = [
    [1.0, 0.0],
    [0.0, 1.0]
]
num_heads = 1

Expected output:

[
    [0.5, 0.5],
    [0.5, 0.5]
]

Your output

Example 2 Not run

Input:

x = [
    [2.0, 4.0]
]
w_q = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_k = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_v = [
    [1.0, 0.0],
    [0.0, 1.0]
]
w_o = [
    [0.0, 1.0],
    [1.0, 0.0]
]
num_heads = 2

Expected output:

[
    [4.0, 2.0]
]

Your output

4 hidden tests also run when you submit. If your code fails one, you'll see that test. Solve the challenge to see them all.

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💡 Kotlin Basics Guide for Beginners

1. Declaring Variables and Constants

Kotlin uses val for immutable variables and var for mutable ones.

val name: String = "Alice"
var age: Int = 30
val pi = 3.14  // type inferred

// Constants
const val MAX_USERS = 100

2. Conditionals (if / when)

Use if expressions or when for multiple branches.

val x = 2
if (x == 1) {
    println("One")
} else if (x == 2) {
    println("Two")
} else {
    println("Other")
}

when (x) {
    1 -> println("One")
    2 -> println("Two")
    else -> println("Other")
}

3. Loops

Use for, while, and do-while for iteration.

for (i in 0..2) {
    println(i)
}

var n = 3
while (n > 0) {
    println(n)
    n--
}

4. Arrays

Kotlin supports arrays using the arrayOf() function.

val numbers = arrayOf(10, 20, 30)
println(numbers[1])

5. List Manipulation

Use mutableListOf for dynamic lists.

val nums = mutableListOf(1, 2, 3)
nums.add(4)
nums.removeAt(0)

for (n in nums) {
    print("$n ")
}

6. Console Input/Output

Use readLine() for input and println() for output.

print("Enter your name: ")
val name = readLine()
println("Hello, $name")

7. Functions

Functions use fun keyword with optional return type.

fun add(a: Int, b: Int): Int {
    return a + b
}

println(add(3, 4))

8. Maps

mutableMapOf stores key-value pairs.

val ages = mutableMapOf("Alice" to 30)
println(ages["Alice"])

9. Exception Handling

Use try, catch, and finally to handle errors.

try {
    val result = 10 / 0
} catch (e: ArithmeticException) {
    println("Error: ${e.message}")
}

10. File I/O

Use File from java.io for file operations.

import java.io.File

File("file.txt").writeText("Hello File")
val text = File("file.txt").readText()
println(text)

11. String Manipulation

Kotlin strings support many methods and interpolation.

val text = "Hello World"
println(text.length)
println(text.substring(0, 5))
println(text.contains("World"))

12. Classes & Objects

Kotlin classes are concise and support default constructors.

class Person(val name: String) {
    fun greet() = println("Hi, I'm $name")
}

val p = Person("Alice")
p.greet()

13. Null Safety

Kotlin distinguishes nullable and non-nullable types. Use ? and ?: for safety.

val name: String? = null
println(name?.length ?: 0)