She is the Goddess of Knowledge Arts and Satwa logic, free from all preconditioning.
White dress to ensure no prior impressions, biases, prejudices, presumptions, halo effects and to clear them all before proceeding to capture truthful and real knowledge based on logic.
Bead chain to ensure that knowledge is retained, chanted and preserved over generations. Note that education in those Sruti days was stored and propagated through repeated chants and by singing. Memory was stored in the throat called ‘Kanthastha’. To remember anything, repeat it loudly 2-5 times with musical rhythm, and it would be stored in the throat. Mind would repeat it subconsciously.
Veena (the guitar) to ensure the knowledge is encased into poetic order and is easily memorized. In addition, Veena represents all kinds of arts, music and creative knowledge.
Book in the hand - obviously to store the knowledge in written form, a repository of knowledge.
Swan, which is said to have the capacity to separate milk from water, representing a quality to separate facts and to discern right and wrong, doable and non-doable, logically.
When a swan moves its off-springs move behind it; indicating a system in which information is imparted from one generation to another and knowledge continues to go forward. Thus, Swan also indicates to carrying forward knowledge through research or through Guru-Shishya parampara (teacher-pupils traditions).