One-hot Encoding · One Category, One Lit Cell

01 · 想象三盏呼叫灯 01 · Imagine three call lights

猫、狗、鸟各按一盏专属灯，能知道谁按了；能知道猫更像狗还是鸟吗？ Cat, dog, and bird each press a dedicated light. You know who pressed it—but can you tell whether cat is closer to dog or bird?

02 点类别，观察哪格亮起 Choose a Category and Watch One Cell Light Up

唯一变量：类别 One variable: category

当前类别 Category

猫 Cat

index 0

亮格 Hot cells

其余全为 0 all others are 0

异类距离 Other-class distance

√2

≈ 1.414

03 三格向量的 4 个零件 Four Parts of the Three-cell Vector

identity, not similarity

离散类别 Discrete category

猫、狗、鸟是互斥标签，不是连续数值。 Cat, dog, and bird are mutually exclusive labels, not continuous numbers.

POSITION

专属位置 Dedicated position

每个类别在向量中占一个固定索引。 Each category owns one fixed vector index.

HOT

唯一的 1 The single 1

所选类别位置亮起，其余位置为 0。 The selected category turns on; every other position stays 0.

DISTANCE

等距 Equal distance

任意两个不同独热向量的距离都为 √2。 Every pair of distinct one-hot vectors is √2 apart.

04 最小机制：索引相同才落在同一点 Minimal Mechanism: Only the Same Index Shares a Point

e _i

onehot(i)_j = { 1 if i=j · 0 otherwise }

不同类别在两个位置上相差 1，所以距离 √(1²+1²) = √2。上方切换任意类别都能验证。 Two distinct categories differ by 1 in two positions, so distance √(1 ²+1² ) = √2. Verify by choosing any category above.

选择类别 Choose category 取得专属索引 i take its index i

第 i 格置 1 Set cell i to 1 其余格清零 zero every other cell

身份可区分 Identity separates 相似性仍未表达 similarity remains absent

05 边界实验：把类别编号当距离会造假 Boundary Test: Treating Category IDs as Distance Invents Order

主动制造失败 Create a failure

把猫=0、狗=1、鸟=2 直接算差 Compute differences from cat=0, dog=1, bird=2

这样会凭空得到“猫到鸟是猫到狗的两倍远”，而编号只用于识别，不承载大小。 This invents “cat is twice as far from bird as from dog,” although IDs identify categories and carry no magnitude.

尚未执行。先确认独热距离全部相同。 Not run yet. First confirm all one-hot distances are equal.

✗ 类别 ID 越接近，类别越相似。 Closer category IDs mean more similar categories. → ✓ ID 是索引，不是量尺。 An ID is an index, not a ruler.

✗ 独热编码能表达猫比鸟更像狗。 One-hot encoding says cat is more like dog than bird. → ✓ 所有不同类别都等距。 All distinct categories are equally distant.

✗ 向量有数字就一定有语义距离。 Numeric vectors always have semantic distance. → ✓ 编码方式决定距离是否有意义。 The encoding determines whether distance is meaningful.

06 · 一句话带走 06 · One line to keep

独热编码用唯一的 1 表示离散类别，能区分身份却让任意两类等距；上方切换类别即可检验。 One-hot encoding identifies a discrete category with one 1, separating identity while making every distinct pair equally distant. Choose categories above to verify.